HomeHome Metamath Proof Explorer
Theorem List (p. 82 of 482)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-30702)
  Hilbert Space Explorer  Hilbert Space Explorer
(30703-32225)
  Users' Mathboxes  Users' Mathboxes
(32226-48151)
 

Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcurry1 8101* Composition with β—‘(2nd β†Ύ ({𝐢} Γ— V)) turns any binary operation 𝐹 with a constant first operand into a function 𝐺 of the second operand only. This transformation is called "currying". (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Dec-2014.)
𝐺 = (𝐹 ∘ β—‘(2nd β†Ύ ({𝐢} Γ— V)))    β‡’   ((𝐹 Fn (𝐴 Γ— 𝐡) ∧ 𝐢 ∈ 𝐴) β†’ 𝐺 = (π‘₯ ∈ 𝐡 ↦ (𝐢𝐹π‘₯)))
 
Theoremcurry1val 8102 The value of a curried function with a constant first argument. (Contributed by NM, 28-Mar-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)
𝐺 = (𝐹 ∘ β—‘(2nd β†Ύ ({𝐢} Γ— V)))    β‡’   ((𝐹 Fn (𝐴 Γ— 𝐡) ∧ 𝐢 ∈ 𝐴) β†’ (πΊβ€˜π·) = (𝐢𝐹𝐷))
 
Theoremcurry1f 8103 Functionality of a curried function with a constant first argument. (Contributed by NM, 29-Mar-2008.)
𝐺 = (𝐹 ∘ β—‘(2nd β†Ύ ({𝐢} Γ— V)))    β‡’   ((𝐹:(𝐴 Γ— 𝐡)⟢𝐷 ∧ 𝐢 ∈ 𝐴) β†’ 𝐺:𝐡⟢𝐷)
 
Theoremcurry2 8104* Composition with β—‘(1st β†Ύ (V Γ— {𝐢})) turns any binary operation 𝐹 with a constant second operand into a function 𝐺 of the first operand only. This transformation is called "currying". (If this becomes frequently used, we can introduce a new notation for the hypothesis.) (Contributed by NM, 16-Dec-2008.)
𝐺 = (𝐹 ∘ β—‘(1st β†Ύ (V Γ— {𝐢})))    β‡’   ((𝐹 Fn (𝐴 Γ— 𝐡) ∧ 𝐢 ∈ 𝐡) β†’ 𝐺 = (π‘₯ ∈ 𝐴 ↦ (π‘₯𝐹𝐢)))
 
Theoremcurry2f 8105 Functionality of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
𝐺 = (𝐹 ∘ β—‘(1st β†Ύ (V Γ— {𝐢})))    β‡’   ((𝐹:(𝐴 Γ— 𝐡)⟢𝐷 ∧ 𝐢 ∈ 𝐡) β†’ 𝐺:𝐴⟢𝐷)
 
Theoremcurry2val 8106 The value of a curried function with a constant second argument. (Contributed by NM, 16-Dec-2008.)
𝐺 = (𝐹 ∘ β—‘(1st β†Ύ (V Γ— {𝐢})))    β‡’   ((𝐹 Fn (𝐴 Γ— 𝐡) ∧ 𝐢 ∈ 𝐡) β†’ (πΊβ€˜π·) = (𝐷𝐹𝐢))
 
Theoremcnvf1olem 8107 Lemma for cnvf1o 8108. (Contributed by Mario Carneiro, 27-Apr-2014.)
((Rel 𝐴 ∧ (𝐡 ∈ 𝐴 ∧ 𝐢 = βˆͺ β—‘{𝐡})) β†’ (𝐢 ∈ ◑𝐴 ∧ 𝐡 = βˆͺ β—‘{𝐢}))
 
Theoremcnvf1o 8108* Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.)
(Rel 𝐴 β†’ (π‘₯ ∈ 𝐴 ↦ βˆͺ β—‘{π‘₯}):𝐴–1-1-onto→◑𝐴)
 
Theoremfparlem1 8109 Lemma for fpar 8113. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
(β—‘(1st β†Ύ (V Γ— V)) β€œ {π‘₯}) = ({π‘₯} Γ— V)
 
Theoremfparlem2 8110 Lemma for fpar 8113. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
(β—‘(2nd β†Ύ (V Γ— V)) β€œ {𝑦}) = (V Γ— {𝑦})
 
Theoremfparlem3 8111* Lemma for fpar 8113. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐹 Fn 𝐴 β†’ (β—‘(1st β†Ύ (V Γ— V)) ∘ (𝐹 ∘ (1st β†Ύ (V Γ— V)))) = βˆͺ π‘₯ ∈ 𝐴 (({π‘₯} Γ— V) Γ— ({(πΉβ€˜π‘₯)} Γ— V)))
 
Theoremfparlem4 8112* Lemma for fpar 8113. (Contributed by NM, 22-Dec-2008.) (Revised by Mario Carneiro, 28-Apr-2015.)
(𝐺 Fn 𝐡 β†’ (β—‘(2nd β†Ύ (V Γ— V)) ∘ (𝐺 ∘ (2nd β†Ύ (V Γ— V)))) = βˆͺ 𝑦 ∈ 𝐡 ((V Γ— {𝑦}) Γ— (V Γ— {(πΊβ€˜π‘¦)})))
 
Theoremfpar 8113* Merge two functions in parallel. Use as the second argument of a composition with a binary operation to build compound functions such as (π‘₯ ∈ (0[,)+∞), 𝑦 ∈ ℝ ↦ ((βˆšβ€˜π‘₯) + (sinβ€˜π‘¦))), see also ex-fpar 30246. (Contributed by NM, 17-Sep-2007.) (Proof shortened by Mario Carneiro, 28-Apr-2015.)
𝐻 = ((β—‘(1st β†Ύ (V Γ— V)) ∘ (𝐹 ∘ (1st β†Ύ (V Γ— V)))) ∩ (β—‘(2nd β†Ύ (V Γ— V)) ∘ (𝐺 ∘ (2nd β†Ύ (V Γ— V)))))    β‡’   ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐡) β†’ 𝐻 = (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐡 ↦ ⟨(πΉβ€˜π‘₯), (πΊβ€˜π‘¦)⟩))
 
Theoremfsplit 8114 A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 8113 in order to build compound functions such as (π‘₯ ∈ (0[,)+∞) ↦ ((βˆšβ€˜π‘₯) + (sinβ€˜π‘₯))). (Contributed by NM, 17-Sep-2007.) Replace use of dfid2 5572 with df-id 5570. (Revised by BJ, 31-Dec-2023.)
β—‘(1st β†Ύ I ) = (π‘₯ ∈ V ↦ ⟨π‘₯, π‘₯⟩)
 
Theoremfsplitfpar 8115* Merge two functions with a common argument in parallel. Combination of fsplit 8114 and fpar 8113. (Contributed by AV, 3-Jan-2024.)
𝐻 = ((β—‘(1st β†Ύ (V Γ— V)) ∘ (𝐹 ∘ (1st β†Ύ (V Γ— V)))) ∩ (β—‘(2nd β†Ύ (V Γ— V)) ∘ (𝐺 ∘ (2nd β†Ύ (V Γ— V)))))    &   π‘† = (β—‘(1st β†Ύ I ) β†Ύ 𝐴)    β‡’   ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) β†’ (𝐻 ∘ 𝑆) = (π‘₯ ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘₯), (πΊβ€˜π‘₯)⟩))
 
Theoremoffsplitfpar 8116 Express the function operation map ∘f by the functions defined in fsplit 8114 and fpar 8113. (Contributed by AV, 4-Jan-2024.)
𝐻 = ((β—‘(1st β†Ύ (V Γ— V)) ∘ (𝐹 ∘ (1st β†Ύ (V Γ— V)))) ∩ (β—‘(2nd β†Ύ (V Γ— V)) ∘ (𝐺 ∘ (2nd β†Ύ (V Γ— V)))))    &   π‘† = (β—‘(1st β†Ύ I ) β†Ύ 𝐴)    β‡’   (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ π‘Š) ∧ ( + Fn 𝐢 ∧ (ran 𝐹 Γ— ran 𝐺) βŠ† 𝐢)) β†’ ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))
 
Theoremf2ndf 8117 The 2nd (second component of an ordered pair) function restricted to a function 𝐹 is a function from 𝐹 into the codomain of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
(𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹⟢𝐡)
 
Theoremfo2ndf 8118 The 2nd (second component of an ordered pair) function restricted to a function 𝐹 is a function from 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
(𝐹:𝐴⟢𝐡 β†’ (2nd β†Ύ 𝐹):𝐹–ontoβ†’ran 𝐹)
 
Theoremf1o2ndf1 8119 The 2nd (second component of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function from 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.)
(𝐹:𝐴–1-1→𝐡 β†’ (2nd β†Ύ 𝐹):𝐹–1-1-ontoβ†’ran 𝐹)
 
Theoremopco1 8120 Value of an operation precomposed with the projection on the first component. (Contributed by Mario Carneiro, 28-May-2014.) Generalize to closed form. (Revised by BJ, 27-Oct-2024.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    β‡’   (πœ‘ β†’ (𝐴(𝐹 ∘ 1st )𝐡) = (πΉβ€˜π΄))
 
Theoremopco2 8121 Value of an operation precomposed with the projection on the second component. (Contributed by BJ, 27-Oct-2024.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝐡 ∈ π‘Š)    β‡’   (πœ‘ β†’ (𝐴(𝐹 ∘ 2nd )𝐡) = (πΉβ€˜π΅))
 
Theoremopco1i 8122 Inference form of opco1 8120. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝐡 ∈ V    &   πΆ ∈ V    β‡’   (𝐡(𝐹 ∘ 1st )𝐢) = (πΉβ€˜π΅)
 
Theoremfrxp 8123* A lexicographical ordering of two well-founded classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) (Proof shortened by Wolf Lammen, 4-Oct-2014.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (𝐴 Γ— 𝐡) ∧ 𝑦 ∈ (𝐴 Γ— 𝐡)) ∧ ((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ ((1st β€˜π‘₯) = (1st β€˜π‘¦) ∧ (2nd β€˜π‘₯)𝑆(2nd β€˜π‘¦))))}    β‡’   ((𝑅 Fr 𝐴 ∧ 𝑆 Fr 𝐡) β†’ 𝑇 Fr (𝐴 Γ— 𝐡))
 
Theoremxporderlem 8124* Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (𝐴 Γ— 𝐡) ∧ 𝑦 ∈ (𝐴 Γ— 𝐡)) ∧ ((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ ((1st β€˜π‘₯) = (1st β€˜π‘¦) ∧ (2nd β€˜π‘₯)𝑆(2nd β€˜π‘¦))))}    β‡’   (βŸ¨π‘Ž, π‘βŸ©π‘‡βŸ¨π‘, π‘‘βŸ© ↔ (((π‘Ž ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐡 ∧ 𝑑 ∈ 𝐡)) ∧ (π‘Žπ‘…π‘ ∨ (π‘Ž = 𝑐 ∧ 𝑏𝑆𝑑))))
 
Theorempoxp 8125* A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (𝐴 Γ— 𝐡) ∧ 𝑦 ∈ (𝐴 Γ— 𝐡)) ∧ ((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ ((1st β€˜π‘₯) = (1st β€˜π‘¦) ∧ (2nd β€˜π‘₯)𝑆(2nd β€˜π‘¦))))}    β‡’   ((𝑅 Po 𝐴 ∧ 𝑆 Po 𝐡) β†’ 𝑇 Po (𝐴 Γ— 𝐡))
 
Theoremsoxp 8126* A lexicographical ordering of two strictly ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (𝐴 Γ— 𝐡) ∧ 𝑦 ∈ (𝐴 Γ— 𝐡)) ∧ ((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ ((1st β€˜π‘₯) = (1st β€˜π‘¦) ∧ (2nd β€˜π‘₯)𝑆(2nd β€˜π‘¦))))}    β‡’   ((𝑅 Or 𝐴 ∧ 𝑆 Or 𝐡) β†’ 𝑇 Or (𝐴 Γ— 𝐡))
 
Theoremwexp 8127* A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (𝐴 Γ— 𝐡) ∧ 𝑦 ∈ (𝐴 Γ— 𝐡)) ∧ ((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ ((1st β€˜π‘₯) = (1st β€˜π‘¦) ∧ (2nd β€˜π‘₯)𝑆(2nd β€˜π‘¦))))}    β‡’   ((𝑅 We 𝐴 ∧ 𝑆 We 𝐡) β†’ 𝑇 We (𝐴 Γ— 𝐡))
 
Theoremfnwelem 8128* Lemma for fnwe 8129. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((πΉβ€˜π‘₯)𝑅(πΉβ€˜π‘¦) ∨ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ∧ π‘₯𝑆𝑦)))}    &   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ 𝑅 We 𝐡)    &   (πœ‘ β†’ 𝑆 We 𝐴)    &   (πœ‘ β†’ (𝐹 β€œ 𝑀) ∈ V)    &   π‘„ = {βŸ¨π‘’, π‘£βŸ© ∣ ((𝑒 ∈ (𝐡 Γ— 𝐴) ∧ 𝑣 ∈ (𝐡 Γ— 𝐴)) ∧ ((1st β€˜π‘’)𝑅(1st β€˜π‘£) ∨ ((1st β€˜π‘’) = (1st β€˜π‘£) ∧ (2nd β€˜π‘’)𝑆(2nd β€˜π‘£))))}    &   πΊ = (𝑧 ∈ 𝐴 ↦ ⟨(πΉβ€˜π‘§), π‘§βŸ©)    β‡’   (πœ‘ β†’ 𝑇 We 𝐴)
 
Theoremfnwe 8129* A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((πΉβ€˜π‘₯)𝑅(πΉβ€˜π‘¦) ∨ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ∧ π‘₯𝑆𝑦)))}    &   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ 𝑅 We 𝐡)    &   (πœ‘ β†’ 𝑆 We 𝐴)    &   (πœ‘ β†’ (𝐹 β€œ 𝑀) ∈ V)    β‡’   (πœ‘ β†’ 𝑇 We 𝐴)
 
Theoremfnse 8130* Condition for the well-order in fnwe 8129 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((πΉβ€˜π‘₯)𝑅(πΉβ€˜π‘¦) ∨ ((πΉβ€˜π‘₯) = (πΉβ€˜π‘¦) ∧ π‘₯𝑆𝑦)))}    &   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ 𝑅 Se 𝐡)    &   (πœ‘ β†’ (◑𝐹 β€œ 𝑀) ∈ V)    β‡’   (πœ‘ β†’ 𝑇 Se 𝐴)
 
Theoremfvproj 8131* Value of a function on ordered pairs with values expressed as ordered pairs. Note that 𝐹 and 𝐺 are the projections of 𝐻 to the first and second coordinate respectively. (Contributed by Thierry Arnoux, 30-Dec-2019.)
𝐻 = (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐡 ↦ ⟨(πΉβ€˜π‘₯), (πΊβ€˜π‘¦)⟩)    &   (πœ‘ β†’ 𝑋 ∈ 𝐴)    &   (πœ‘ β†’ π‘Œ ∈ 𝐡)    β‡’   (πœ‘ β†’ (π»β€˜βŸ¨π‘‹, π‘ŒβŸ©) = ⟨(πΉβ€˜π‘‹), (πΊβ€˜π‘Œ)⟩)
 
Theoremfimaproj 8132* Image of a cartesian product for a function on ordered pairs with values expressed as ordered pairs. Note that 𝐹 and 𝐺 are the projections of 𝐻 to the first and second coordinate respectively. (Contributed by Thierry Arnoux, 30-Dec-2019.)
𝐻 = (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐡 ↦ ⟨(πΉβ€˜π‘₯), (πΊβ€˜π‘¦)⟩)    &   (πœ‘ β†’ 𝐹 Fn 𝐴)    &   (πœ‘ β†’ 𝐺 Fn 𝐡)    &   (πœ‘ β†’ 𝑋 βŠ† 𝐴)    &   (πœ‘ β†’ π‘Œ βŠ† 𝐡)    β‡’   (πœ‘ β†’ (𝐻 β€œ (𝑋 Γ— π‘Œ)) = ((𝐹 β€œ 𝑋) Γ— (𝐺 β€œ π‘Œ)))
 
Theoremralxpes 8133* A version of ralxp 5838 with explicit substitution. (Contributed by Scott Fenton, 21-Aug-2024.)
(βˆ€π‘₯ ∈ (𝐴 Γ— 𝐡)[(1st β€˜π‘₯) / 𝑦][(2nd β€˜π‘₯) / 𝑧]πœ‘ ↔ βˆ€π‘¦ ∈ 𝐴 βˆ€π‘§ ∈ 𝐡 πœ‘)
 
Theoremralxp3f 8134* Restricted for all over a triple Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)
β„²π‘¦πœ‘    &   β„²π‘§πœ‘    &   β„²π‘€πœ‘    &   β„²π‘₯πœ“    &   (π‘₯ = βŸ¨π‘¦, 𝑧, π‘€βŸ© β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢)πœ‘ ↔ βˆ€π‘¦ ∈ 𝐴 βˆ€π‘§ ∈ 𝐡 βˆ€π‘€ ∈ 𝐢 πœ“)
 
Theoremralxp3 8135* Restricted for all over a triple Cartesian product. (Contributed by Scott Fenton, 2-Feb-2025.)
(π‘₯ = βŸ¨π‘¦, 𝑧, π‘€βŸ© β†’ (πœ‘ ↔ πœ“))    β‡’   (βˆ€π‘₯ ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢)πœ‘ ↔ βˆ€π‘¦ ∈ 𝐴 βˆ€π‘§ ∈ 𝐡 βˆ€π‘€ ∈ 𝐢 πœ“)
 
Theoremralxp3es 8136* Restricted for-all over a triple Cartesian product with explicit substitution. (Contributed by Scott Fenton, 22-Aug-2024.)
(βˆ€π‘₯ ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢)[(1st β€˜(1st β€˜π‘₯)) / 𝑦][(2nd β€˜(1st β€˜π‘₯)) / 𝑧][(2nd β€˜π‘₯) / 𝑀]πœ‘ ↔ βˆ€π‘¦ ∈ 𝐴 βˆ€π‘§ ∈ 𝐡 βˆ€π‘€ ∈ 𝐢 πœ‘)
 
2.4.9  Induction on Cartesian products
 
Theoremfrpoins3xpg 8137* Special case of founded partial induction over a Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)
((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡) β†’ (βˆ€π‘§βˆ€π‘€(βŸ¨π‘§, π‘€βŸ© ∈ Pred(𝑅, (𝐴 Γ— 𝐡), ⟨π‘₯, π‘¦βŸ©) β†’ πœ’) β†’ πœ‘))    &   (π‘₯ = 𝑧 β†’ (πœ‘ ↔ πœ“))    &   (𝑦 = 𝑀 β†’ (πœ“ ↔ πœ’))    &   (π‘₯ = 𝑋 β†’ (πœ‘ ↔ πœƒ))    &   (𝑦 = π‘Œ β†’ (πœƒ ↔ 𝜏))    β‡’   (((𝑅 Fr (𝐴 Γ— 𝐡) ∧ 𝑅 Po (𝐴 Γ— 𝐡) ∧ 𝑅 Se (𝐴 Γ— 𝐡)) ∧ (𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐡)) β†’ 𝜏)
 
Theoremfrpoins3xp3g 8138* Special case of founded partial recursion over a triple Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)
((π‘₯ ∈ 𝐴 ∧ 𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐢) β†’ (βˆ€π‘€βˆ€π‘‘βˆ€π‘’(βŸ¨π‘€, 𝑑, π‘’βŸ© ∈ Pred(𝑅, ((𝐴 Γ— 𝐡) Γ— 𝐢), ⟨π‘₯, 𝑦, π‘§βŸ©) β†’ πœƒ) β†’ πœ‘))    &   (π‘₯ = 𝑀 β†’ (πœ‘ ↔ πœ“))    &   (𝑦 = 𝑑 β†’ (πœ“ ↔ πœ’))    &   (𝑧 = 𝑒 β†’ (πœ’ ↔ πœƒ))    &   (π‘₯ = 𝑋 β†’ (πœ‘ ↔ 𝜏))    &   (𝑦 = π‘Œ β†’ (𝜏 ↔ πœ‚))    &   (𝑧 = 𝑍 β†’ (πœ‚ ↔ 𝜁))    β‡’   (((𝑅 Fr ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ 𝑅 Po ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ 𝑅 Se ((𝐴 Γ— 𝐡) Γ— 𝐢)) ∧ (𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐢)) β†’ 𝜁)
 
2.4.10  Ordering on Cartesian products
 
Theoremxpord2lem 8139* Lemma for Cartesian product ordering. Calculate the value of the Cartesian product relation. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (𝐴 Γ— 𝐡) ∧ 𝑦 ∈ (𝐴 Γ— 𝐡) ∧ (((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ (1st β€˜π‘₯) = (1st β€˜π‘¦)) ∧ ((2nd β€˜π‘₯)𝑆(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦)) ∧ π‘₯ β‰  𝑦))}    β‡’   (βŸ¨π‘Ž, π‘βŸ©π‘‡βŸ¨π‘, π‘‘βŸ© ↔ ((π‘Ž ∈ 𝐴 ∧ 𝑏 ∈ 𝐡) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐡) ∧ ((π‘Žπ‘…π‘ ∨ π‘Ž = 𝑐) ∧ (𝑏𝑆𝑑 ∨ 𝑏 = 𝑑) ∧ (π‘Ž β‰  𝑐 ∨ 𝑏 β‰  𝑑))))
 
Theorempoxp2 8140* Another way of partially ordering a Cartesian product of two classes. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (𝐴 Γ— 𝐡) ∧ 𝑦 ∈ (𝐴 Γ— 𝐡) ∧ (((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ (1st β€˜π‘₯) = (1st β€˜π‘¦)) ∧ ((2nd β€˜π‘₯)𝑆(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦)) ∧ π‘₯ β‰  𝑦))}    &   (πœ‘ β†’ 𝑅 Po 𝐴)    &   (πœ‘ β†’ 𝑆 Po 𝐡)    β‡’   (πœ‘ β†’ 𝑇 Po (𝐴 Γ— 𝐡))
 
Theoremfrxp2 8141* Another way of giving a well-founded order to a Cartesian product of two classes. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (𝐴 Γ— 𝐡) ∧ 𝑦 ∈ (𝐴 Γ— 𝐡) ∧ (((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ (1st β€˜π‘₯) = (1st β€˜π‘¦)) ∧ ((2nd β€˜π‘₯)𝑆(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦)) ∧ π‘₯ β‰  𝑦))}    &   (πœ‘ β†’ 𝑅 Fr 𝐴)    &   (πœ‘ β†’ 𝑆 Fr 𝐡)    β‡’   (πœ‘ β†’ 𝑇 Fr (𝐴 Γ— 𝐡))
 
Theoremxpord2pred 8142* Calculate the predecessor class in frxp2 8141. (Contributed by Scott Fenton, 22-Aug-2024.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (𝐴 Γ— 𝐡) ∧ 𝑦 ∈ (𝐴 Γ— 𝐡) ∧ (((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ (1st β€˜π‘₯) = (1st β€˜π‘¦)) ∧ ((2nd β€˜π‘₯)𝑆(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦)) ∧ π‘₯ β‰  𝑦))}    β‡’   ((𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐡) β†’ Pred(𝑇, (𝐴 Γ— 𝐡), βŸ¨π‘‹, π‘ŒβŸ©) = (((Pred(𝑅, 𝐴, 𝑋) βˆͺ {𝑋}) Γ— (Pred(𝑆, 𝐡, π‘Œ) βˆͺ {π‘Œ})) βˆ– {βŸ¨π‘‹, π‘ŒβŸ©}))
 
Theoremsexp2 8143* Condition for the relation in frxp2 8141 to be set-like. (Contributed by Scott Fenton, 19-Aug-2024.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (𝐴 Γ— 𝐡) ∧ 𝑦 ∈ (𝐴 Γ— 𝐡) ∧ (((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ (1st β€˜π‘₯) = (1st β€˜π‘¦)) ∧ ((2nd β€˜π‘₯)𝑆(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦)) ∧ π‘₯ β‰  𝑦))}    &   (πœ‘ β†’ 𝑅 Se 𝐴)    &   (πœ‘ β†’ 𝑆 Se 𝐡)    β‡’   (πœ‘ β†’ 𝑇 Se (𝐴 Γ— 𝐡))
 
Theoremxpord2indlem 8144* Induction over the Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 22-Aug-2024.)
𝑇 = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ (𝐴 Γ— 𝐡) ∧ 𝑦 ∈ (𝐴 Γ— 𝐡) ∧ (((1st β€˜π‘₯)𝑅(1st β€˜π‘¦) ∨ (1st β€˜π‘₯) = (1st β€˜π‘¦)) ∧ ((2nd β€˜π‘₯)𝑆(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦)) ∧ π‘₯ β‰  𝑦))}    &   π‘… Fr 𝐴    &   π‘… Po 𝐴    &   π‘… Se 𝐴    &   π‘† Fr 𝐡    &   π‘† Po 𝐡    &   π‘† Se 𝐡    &   (π‘Ž = 𝑐 β†’ (πœ‘ ↔ πœ“))    &   (𝑏 = 𝑑 β†’ (πœ“ ↔ πœ’))    &   (π‘Ž = 𝑐 β†’ (πœƒ ↔ πœ’))    &   (π‘Ž = 𝑋 β†’ (πœ‘ ↔ 𝜏))    &   (𝑏 = π‘Œ β†’ (𝜏 ↔ πœ‚))    &   ((π‘Ž ∈ 𝐴 ∧ 𝑏 ∈ 𝐡) β†’ ((βˆ€π‘ ∈ Pred (𝑅, 𝐴, π‘Ž)βˆ€π‘‘ ∈ Pred (𝑆, 𝐡, 𝑏)πœ’ ∧ βˆ€π‘ ∈ Pred (𝑅, 𝐴, π‘Ž)πœ“ ∧ βˆ€π‘‘ ∈ Pred (𝑆, 𝐡, 𝑏)πœƒ) β†’ πœ‘))    β‡’   ((𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐡) β†’ πœ‚)
 
Theoremxpord2ind 8145* Induction over the Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 22-Aug-2024.)
𝑅 Fr 𝐴    &   π‘… Po 𝐴    &   π‘… Se 𝐴    &   π‘† Fr 𝐡    &   π‘† Po 𝐡    &   π‘† Se 𝐡    &   (π‘Ž = 𝑐 β†’ (πœ‘ ↔ πœ“))    &   (𝑏 = 𝑑 β†’ (πœ“ ↔ πœ’))    &   (π‘Ž = 𝑐 β†’ (πœƒ ↔ πœ’))    &   (π‘Ž = 𝑋 β†’ (πœ‘ ↔ 𝜏))    &   (𝑏 = π‘Œ β†’ (𝜏 ↔ πœ‚))    &   ((π‘Ž ∈ 𝐴 ∧ 𝑏 ∈ 𝐡) β†’ ((βˆ€π‘ ∈ Pred (𝑅, 𝐴, π‘Ž)βˆ€π‘‘ ∈ Pred (𝑆, 𝐡, 𝑏)πœ’ ∧ βˆ€π‘ ∈ Pred (𝑅, 𝐴, π‘Ž)πœ“ ∧ βˆ€π‘‘ ∈ Pred (𝑆, 𝐡, 𝑏)πœƒ) β†’ πœ‘))    β‡’   ((𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐡) β†’ πœ‚)
 
Theoremxpord3lem 8146* Lemma for triple ordering. Calculate the value of the relation. (Contributed by Scott Fenton, 21-Aug-2024.)
π‘ˆ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ 𝑦 ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ ((((1st β€˜(1st β€˜π‘₯))𝑅(1st β€˜(1st β€˜π‘¦)) ∨ (1st β€˜(1st β€˜π‘₯)) = (1st β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜(1st β€˜π‘₯))𝑆(2nd β€˜(1st β€˜π‘¦)) ∨ (2nd β€˜(1st β€˜π‘₯)) = (2nd β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜π‘₯)𝑇(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦))) ∧ π‘₯ β‰  𝑦))}    β‡’   (βŸ¨π‘Ž, 𝑏, π‘βŸ©π‘ˆβŸ¨π‘‘, 𝑒, π‘“βŸ© ↔ ((π‘Ž ∈ 𝐴 ∧ 𝑏 ∈ 𝐡 ∧ 𝑐 ∈ 𝐢) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐢) ∧ (((π‘Žπ‘…π‘‘ ∨ π‘Ž = 𝑑) ∧ (𝑏𝑆𝑒 ∨ 𝑏 = 𝑒) ∧ (𝑐𝑇𝑓 ∨ 𝑐 = 𝑓)) ∧ (π‘Ž β‰  𝑑 ∨ 𝑏 β‰  𝑒 ∨ 𝑐 β‰  𝑓))))
 
Theorempoxp3 8147* Triple Cartesian product partial ordering. (Contributed by Scott Fenton, 21-Aug-2024.)
π‘ˆ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ 𝑦 ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ ((((1st β€˜(1st β€˜π‘₯))𝑅(1st β€˜(1st β€˜π‘¦)) ∨ (1st β€˜(1st β€˜π‘₯)) = (1st β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜(1st β€˜π‘₯))𝑆(2nd β€˜(1st β€˜π‘¦)) ∨ (2nd β€˜(1st β€˜π‘₯)) = (2nd β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜π‘₯)𝑇(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦))) ∧ π‘₯ β‰  𝑦))}    &   (πœ‘ β†’ 𝑅 Po 𝐴)    &   (πœ‘ β†’ 𝑆 Po 𝐡)    &   (πœ‘ β†’ 𝑇 Po 𝐢)    β‡’   (πœ‘ β†’ π‘ˆ Po ((𝐴 Γ— 𝐡) Γ— 𝐢))
 
Theoremfrxp3 8148* Give well-foundedness over a triple Cartesian product. (Contributed by Scott Fenton, 21-Aug-2024.)
π‘ˆ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ 𝑦 ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ ((((1st β€˜(1st β€˜π‘₯))𝑅(1st β€˜(1st β€˜π‘¦)) ∨ (1st β€˜(1st β€˜π‘₯)) = (1st β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜(1st β€˜π‘₯))𝑆(2nd β€˜(1st β€˜π‘¦)) ∨ (2nd β€˜(1st β€˜π‘₯)) = (2nd β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜π‘₯)𝑇(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦))) ∧ π‘₯ β‰  𝑦))}    &   (πœ‘ β†’ 𝑅 Fr 𝐴)    &   (πœ‘ β†’ 𝑆 Fr 𝐡)    &   (πœ‘ β†’ 𝑇 Fr 𝐢)    β‡’   (πœ‘ β†’ π‘ˆ Fr ((𝐴 Γ— 𝐡) Γ— 𝐢))
 
Theoremxpord3pred 8149* Calculate the predecsessor class for the triple order. (Contributed by Scott Fenton, 31-Jan-2025.)
π‘ˆ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ 𝑦 ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ ((((1st β€˜(1st β€˜π‘₯))𝑅(1st β€˜(1st β€˜π‘¦)) ∨ (1st β€˜(1st β€˜π‘₯)) = (1st β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜(1st β€˜π‘₯))𝑆(2nd β€˜(1st β€˜π‘¦)) ∨ (2nd β€˜(1st β€˜π‘₯)) = (2nd β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜π‘₯)𝑇(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦))) ∧ π‘₯ β‰  𝑦))}    β‡’   ((𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐢) β†’ Pred(π‘ˆ, ((𝐴 Γ— 𝐡) Γ— 𝐢), βŸ¨π‘‹, π‘Œ, π‘βŸ©) = ((((Pred(𝑅, 𝐴, 𝑋) βˆͺ {𝑋}) Γ— (Pred(𝑆, 𝐡, π‘Œ) βˆͺ {π‘Œ})) Γ— (Pred(𝑇, 𝐢, 𝑍) βˆͺ {𝑍})) βˆ– {βŸ¨π‘‹, π‘Œ, π‘βŸ©}))
 
Theoremsexp3 8150* Show that the triple order is set-like. (Contributed by Scott Fenton, 21-Aug-2024.)
π‘ˆ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ 𝑦 ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ ((((1st β€˜(1st β€˜π‘₯))𝑅(1st β€˜(1st β€˜π‘¦)) ∨ (1st β€˜(1st β€˜π‘₯)) = (1st β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜(1st β€˜π‘₯))𝑆(2nd β€˜(1st β€˜π‘¦)) ∨ (2nd β€˜(1st β€˜π‘₯)) = (2nd β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜π‘₯)𝑇(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦))) ∧ π‘₯ β‰  𝑦))}    &   (πœ‘ β†’ 𝑅 Se 𝐴)    &   (πœ‘ β†’ 𝑆 Se 𝐡)    &   (πœ‘ β†’ 𝑇 Se 𝐢)    β‡’   (πœ‘ β†’ π‘ˆ Se ((𝐴 Γ— 𝐡) Γ— 𝐢))
 
Theoremxpord3inddlem 8151* Induction over the triple Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 2-Feb-2025.)
π‘ˆ = {⟨π‘₯, π‘¦βŸ© ∣ (π‘₯ ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ 𝑦 ∈ ((𝐴 Γ— 𝐡) Γ— 𝐢) ∧ ((((1st β€˜(1st β€˜π‘₯))𝑅(1st β€˜(1st β€˜π‘¦)) ∨ (1st β€˜(1st β€˜π‘₯)) = (1st β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜(1st β€˜π‘₯))𝑆(2nd β€˜(1st β€˜π‘¦)) ∨ (2nd β€˜(1st β€˜π‘₯)) = (2nd β€˜(1st β€˜π‘¦))) ∧ ((2nd β€˜π‘₯)𝑇(2nd β€˜π‘¦) ∨ (2nd β€˜π‘₯) = (2nd β€˜π‘¦))) ∧ π‘₯ β‰  𝑦))}    &   (πœ… β†’ 𝑋 ∈ 𝐴)    &   (πœ… β†’ π‘Œ ∈ 𝐡)    &   (πœ… β†’ 𝑍 ∈ 𝐢)    &   (πœ… β†’ 𝑅 Fr 𝐴)    &   (πœ… β†’ 𝑅 Po 𝐴)    &   (πœ… β†’ 𝑅 Se 𝐴)    &   (πœ… β†’ 𝑆 Fr 𝐡)    &   (πœ… β†’ 𝑆 Po 𝐡)    &   (πœ… β†’ 𝑆 Se 𝐡)    &   (πœ… β†’ 𝑇 Fr 𝐢)    &   (πœ… β†’ 𝑇 Po 𝐢)    &   (πœ… β†’ 𝑇 Se 𝐢)    &   (π‘Ž = 𝑑 β†’ (πœ‘ ↔ πœ“))    &   (𝑏 = 𝑒 β†’ (πœ“ ↔ πœ’))    &   (𝑐 = 𝑓 β†’ (πœ’ ↔ πœƒ))    &   (π‘Ž = 𝑑 β†’ (𝜏 ↔ πœƒ))    &   (𝑏 = 𝑒 β†’ (πœ‚ ↔ 𝜏))    &   (𝑏 = 𝑒 β†’ (𝜁 ↔ πœƒ))    &   (𝑐 = 𝑓 β†’ (𝜎 ↔ 𝜏))    &   (π‘Ž = 𝑋 β†’ (πœ‘ ↔ 𝜌))    &   (𝑏 = π‘Œ β†’ (𝜌 ↔ πœ‡))    &   (𝑐 = 𝑍 β†’ (πœ‡ ↔ πœ†))    &   ((πœ… ∧ (π‘Ž ∈ 𝐴 ∧ 𝑏 ∈ 𝐡 ∧ 𝑐 ∈ 𝐢)) β†’ (((βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)πœƒ ∧ βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)πœ’ ∧ βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)𝜁) ∧ (βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)πœ“ ∧ βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)𝜏 ∧ βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)𝜎) ∧ βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)πœ‚) β†’ πœ‘))    β‡’   (πœ… β†’ πœ†)
 
Theoremxpord3indd 8152* Induction over the triple Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 2-Feb-2025.)
(πœ… β†’ 𝑋 ∈ 𝐴)    &   (πœ… β†’ π‘Œ ∈ 𝐡)    &   (πœ… β†’ 𝑍 ∈ 𝐢)    &   (πœ… β†’ 𝑅 Fr 𝐴)    &   (πœ… β†’ 𝑅 Po 𝐴)    &   (πœ… β†’ 𝑅 Se 𝐴)    &   (πœ… β†’ 𝑆 Fr 𝐡)    &   (πœ… β†’ 𝑆 Po 𝐡)    &   (πœ… β†’ 𝑆 Se 𝐡)    &   (πœ… β†’ 𝑇 Fr 𝐢)    &   (πœ… β†’ 𝑇 Po 𝐢)    &   (πœ… β†’ 𝑇 Se 𝐢)    &   (π‘Ž = 𝑑 β†’ (πœ‘ ↔ πœ“))    &   (𝑏 = 𝑒 β†’ (πœ“ ↔ πœ’))    &   (𝑐 = 𝑓 β†’ (πœ’ ↔ πœƒ))    &   (π‘Ž = 𝑑 β†’ (𝜏 ↔ πœƒ))    &   (𝑏 = 𝑒 β†’ (πœ‚ ↔ 𝜏))    &   (𝑏 = 𝑒 β†’ (𝜁 ↔ πœƒ))    &   (𝑐 = 𝑓 β†’ (𝜎 ↔ 𝜏))    &   (π‘Ž = 𝑋 β†’ (πœ‘ ↔ 𝜌))    &   (𝑏 = π‘Œ β†’ (𝜌 ↔ πœ‡))    &   (𝑐 = 𝑍 β†’ (πœ‡ ↔ πœ†))    &   ((πœ… ∧ (π‘Ž ∈ 𝐴 ∧ 𝑏 ∈ 𝐡 ∧ 𝑐 ∈ 𝐢)) β†’ (((βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)πœƒ ∧ βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)πœ’ ∧ βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)𝜁) ∧ (βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)πœ“ ∧ βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)𝜏 ∧ βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)𝜎) ∧ βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)πœ‚) β†’ πœ‘))    β‡’   (πœ… β†’ πœ†)
 
Theoremxpord3ind 8153* Induction over the triple Cartesian product ordering. Note that the substitutions cover all possible cases of membership in the predecessor class. (Contributed by Scott Fenton, 4-Sep-2024.)
𝑅 Fr 𝐴    &   π‘… Po 𝐴    &   π‘… Se 𝐴    &   π‘† Fr 𝐡    &   π‘† Po 𝐡    &   π‘† Se 𝐡    &   π‘‡ Fr 𝐢    &   π‘‡ Po 𝐢    &   π‘‡ Se 𝐢    &   (π‘Ž = 𝑑 β†’ (πœ‘ ↔ πœ“))    &   (𝑏 = 𝑒 β†’ (πœ“ ↔ πœ’))    &   (𝑐 = 𝑓 β†’ (πœ’ ↔ πœƒ))    &   (π‘Ž = 𝑑 β†’ (𝜏 ↔ πœƒ))    &   (𝑏 = 𝑒 β†’ (πœ‚ ↔ 𝜏))    &   (𝑏 = 𝑒 β†’ (𝜁 ↔ πœƒ))    &   (𝑐 = 𝑓 β†’ (𝜎 ↔ 𝜏))    &   (π‘Ž = 𝑋 β†’ (πœ‘ ↔ 𝜌))    &   (𝑏 = π‘Œ β†’ (𝜌 ↔ πœ‡))    &   (𝑐 = 𝑍 β†’ (πœ‡ ↔ πœ†))    &   ((π‘Ž ∈ 𝐴 ∧ 𝑏 ∈ 𝐡 ∧ 𝑐 ∈ 𝐢) β†’ (((βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)πœƒ ∧ βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)πœ’ ∧ βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)𝜁) ∧ (βˆ€π‘‘ ∈ Pred (𝑅, 𝐴, π‘Ž)πœ“ ∧ βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)𝜏 ∧ βˆ€π‘’ ∈ Pred (𝑆, 𝐡, 𝑏)𝜎) ∧ βˆ€π‘“ ∈ Pred (𝑇, 𝐢, 𝑐)πœ‚) β†’ πœ‘))    β‡’   ((𝑋 ∈ 𝐴 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐢) β†’ πœ†)
 
2.4.11  Ordering Ordinal Sequences
 
Theoremorderseqlem 8154* Lemma for poseq 8155 and soseq 8156. The function value of a sequence is either in 𝐴 or null. (Contributed by Scott Fenton, 8-Jun-2011.)
𝐹 = {𝑓 ∣ βˆƒπ‘₯ ∈ On 𝑓:π‘₯⟢𝐴}    β‡’   (𝐺 ∈ 𝐹 β†’ (πΊβ€˜π‘‹) ∈ (𝐴 βˆͺ {βˆ…}))
 
Theoremposeq 8155* A partial ordering of ordinal sequences. (Contributed by Scott Fenton, 8-Jun-2011.)
𝑅 Po (𝐴 βˆͺ {βˆ…})    &   πΉ = {𝑓 ∣ βˆƒπ‘₯ ∈ On 𝑓:π‘₯⟢𝐴}    &   π‘† = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹) ∧ βˆƒπ‘₯ ∈ On (βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (π‘”β€˜π‘¦) ∧ (π‘“β€˜π‘₯)𝑅(π‘”β€˜π‘₯)))}    β‡’   π‘† Po 𝐹
 
Theoremsoseq 8156* A linear ordering of ordinal sequences. (Contributed by Scott Fenton, 8-Jun-2011.)
𝑅 Or (𝐴 βˆͺ {βˆ…})    &   πΉ = {𝑓 ∣ βˆƒπ‘₯ ∈ On 𝑓:π‘₯⟢𝐴}    &   π‘† = {βŸ¨π‘“, π‘”βŸ© ∣ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹) ∧ βˆƒπ‘₯ ∈ On (βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) = (π‘”β€˜π‘¦) ∧ (π‘“β€˜π‘₯)𝑅(π‘”β€˜π‘₯)))}    &    Β¬ βˆ… ∈ 𝐴    β‡’   π‘† Or 𝐹
 
2.4.12  The support of functions

In this section, the support of functions is defined and corresponding theorems are provided. Since basic properties (see suppval 8159) are based on the Axiom of Union (usage of dmexg 7901), these definition and theorems cannot be provided earlier. Until April 2019, the support of a function was represented by the expression (◑𝑅 β€œ (V βˆ– {𝑍})) (see suppimacnv 8170). The theorems which are based on this representation and which are provided in previous sections could be moved into this section to have all related theorems in one section, although they do not depend on the Axiom of Union. This was possible because they are not used before. The current theorems differ from the original ones by requiring that the classes representing the "function" (or its "domain") and the "zero element" are sets. Actually, this does not cause any problem (until now).

 
Syntaxcsupp 8157 Extend class definition to include the support of functions.
class supp
 
Definitiondf-supp 8158* Define the support of a function against a "zero" value. According to Wikipedia ("Support (mathematics)", 31-Mar-2019, https://en.wikipedia.org/wiki/Support_(mathematics)) "In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero." and "The notion of support also extends in a natural way to functions taking values in more general sets than R [the real numbers] and to other objects." The following definition allows for such extensions, being applicable for any sets (which usually are functions) and any element (even not necessarily from the range of the function) regarded as "zero". (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
supp = (π‘₯ ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom π‘₯ ∣ (π‘₯ β€œ {𝑖}) β‰  {𝑧}})
 
Theoremsuppval 8159* The value of the operation constructing the support of a function. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)
((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 β€œ {𝑖}) β‰  {𝑍}})
 
Theoremsupp0prc 8160 The support of a class is empty if either the class or the "zero" is a proper class. (Contributed by AV, 28-May-2019.)
(Β¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) β†’ (𝑋 supp 𝑍) = βˆ…)
 
Theoremsuppvalbr 8161* The value of the operation constructing the support of a function expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (𝑅 supp 𝑍) = {π‘₯ ∣ (βˆƒπ‘¦ π‘₯𝑅𝑦 ∧ βˆƒπ‘¦(π‘₯𝑅𝑦 ↔ 𝑦 β‰  𝑍))})
 
Theoremsupp0 8162 The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
(𝑍 ∈ π‘Š β†’ (βˆ… supp 𝑍) = βˆ…)
 
Theoremsuppval1 8163* The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.)
((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (π‘‹β€˜π‘–) β‰  𝑍})
 
Theoremsuppvalfng 8164* The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 8165 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5279. (Contributed by SN, 5-Aug-2024.)
((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (πΉβ€˜π‘–) β‰  𝑍})
 
Theoremsuppvalfn 8165* The value of the operation constructing the support of a function with a given domain. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 22-Apr-2019.)
((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (πΉβ€˜π‘–) β‰  𝑍})
 
Theoremelsuppfng 8166 An element of the support of a function with a given domain. This version of elsuppfn 8167 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5279. (Contributed by SN, 5-Aug-2024.)
((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆 ∈ 𝑋 ∧ (πΉβ€˜π‘†) β‰  𝑍)))
 
Theoremelsuppfn 8167 An element of the support of a function with a given domain. (Contributed by AV, 27-May-2019.)
((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆 ∈ 𝑋 ∧ (πΉβ€˜π‘†) β‰  𝑍)))
 
Theoremcnvimadfsn 8168* The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
(◑𝑅 β€œ (V βˆ– {𝑍})) = {π‘₯ ∣ βˆƒπ‘¦(π‘₯𝑅𝑦 ∧ 𝑦 β‰  𝑍)}
 
Theoremsuppimacnvss 8169 The support of functions "defined" by inverse images is a subset of the support defined by df-supp 8158. (Contributed by AV, 7-Apr-2019.)
((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (◑𝑅 β€œ (V βˆ– {𝑍})) βŠ† (𝑅 supp 𝑍))
 
Theoremsuppimacnv 8170 Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)
((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (𝑅 supp 𝑍) = (◑𝑅 β€œ (V βˆ– {𝑍})))
 
Theoremfsuppeq 8171 Two ways of writing the support of a function with known codomain. (Contributed by Stefan O'Rear, 9-Jul-2015.) (Revised by AV, 7-Jul-2019.)
((𝐼 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (𝐹:πΌβŸΆπ‘† β†’ (𝐹 supp 𝑍) = (◑𝐹 β€œ (𝑆 βˆ– {𝑍}))))
 
Theoremfsuppeqg 8172 Version of fsuppeq 8171 avoiding ax-rep 5279 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 30-Jul-2024.)
((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (𝐹:πΌβŸΆπ‘† β†’ (𝐹 supp 𝑍) = (◑𝐹 β€œ (𝑆 βˆ– {𝑍}))))
 
Theoremsuppssdm 8173 The support of a function is a subset of the function's domain. (Contributed by AV, 30-May-2019.)
(𝐹 supp 𝑍) βŠ† dom 𝐹
 
Theoremsuppsnop 8174 The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.)
𝐹 = {βŸ¨π‘‹, π‘ŒβŸ©}    β‡’   ((𝑋 ∈ 𝑉 ∧ π‘Œ ∈ π‘Š ∧ 𝑍 ∈ π‘ˆ) β†’ (𝐹 supp 𝑍) = if(π‘Œ = 𝑍, βˆ…, {𝑋}))
 
Theoremsnopsuppss 8175 The support of a singleton containing an ordered pair is a subset of the singleton containing the first element of the ordered pair, i.e. it is empty or the singleton itself. (Contributed by AV, 19-Jul-2019.)
({βŸ¨π‘‹, π‘ŒβŸ©} supp 𝑍) βŠ† {𝑋}
 
Theoremfvn0elsupp 8176 If the function value for a given argument is not empty, the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 2-Jul-2019.) (Revised by AV, 4-Apr-2020.)
(((𝐡 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡) ∧ (𝐺 Fn 𝐡 ∧ (πΊβ€˜π‘‹) β‰  βˆ…)) β†’ 𝑋 ∈ (𝐺 supp βˆ…))
 
Theoremfvn0elsuppb 8177 The function value for a given argument is not empty iff the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 4-Apr-2020.)
((𝐡 ∈ 𝑉 ∧ 𝑋 ∈ 𝐡 ∧ 𝐺 Fn 𝐡) β†’ ((πΊβ€˜π‘‹) β‰  βˆ… ↔ 𝑋 ∈ (𝐺 supp βˆ…)))
 
Theoremrexsupp 8178* Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by AV, 27-May-2019.)
((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (βˆƒπ‘₯ ∈ (𝐹 supp 𝑍)πœ‘ ↔ βˆƒπ‘₯ ∈ 𝑋 ((πΉβ€˜π‘₯) β‰  𝑍 ∧ πœ‘)))
 
Theoremressuppss 8179 The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ ((𝐹 β†Ύ 𝐡) supp 𝑍) βŠ† (𝐹 supp 𝑍))
 
Theoremsuppun 8180 The support of a class/function is a subset of the support of the union of this class/function with another class/function. (Contributed by AV, 4-Jun-2019.)
(πœ‘ β†’ 𝐺 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝐹 supp 𝑍) βŠ† ((𝐹 βˆͺ 𝐺) supp 𝑍))
 
Theoremressuppssdif 8181 The support of the restriction of a function is a subset of the support of the function itself. (Contributed by AV, 22-Apr-2019.)
((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š) β†’ (𝐹 supp 𝑍) βŠ† (((𝐹 β†Ύ 𝐡) supp 𝑍) βˆͺ (dom 𝐹 βˆ– 𝐡)))
 
Theoremmptsuppdifd 8182* The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 ∈ π‘Š)    β‡’   (πœ‘ β†’ (𝐹 supp 𝑍) = {π‘₯ ∈ 𝐴 ∣ 𝐡 ∈ (V βˆ– {𝑍})})
 
Theoremmptsuppd 8183* The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.)
𝐹 = (π‘₯ ∈ 𝐴 ↦ 𝐡)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 ∈ π‘Š)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐡 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (𝐹 supp 𝑍) = {π‘₯ ∈ 𝐴 ∣ 𝐡 β‰  𝑍})
 
Theoremextmptsuppeq 8184* The support of an extended function is the same as the original. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 30-Jun-2019.)
(πœ‘ β†’ 𝐡 ∈ π‘Š)    &   (πœ‘ β†’ 𝐴 βŠ† 𝐡)    &   ((πœ‘ ∧ 𝑛 ∈ (𝐡 βˆ– 𝐴)) β†’ 𝑋 = 𝑍)    β‡’   (πœ‘ β†’ ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐡 ↦ 𝑋) supp 𝑍))
 
Theoremsuppfnss 8185* The support of a function which has the same zero values (in its domain) as another function is a subset of the support of this other function. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 6-Jun-2022.)
(((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐡) ∧ (𝐴 βŠ† 𝐡 ∧ 𝐡 ∈ 𝑉 ∧ 𝑍 ∈ π‘Š)) β†’ (βˆ€π‘₯ ∈ 𝐴 ((πΊβ€˜π‘₯) = 𝑍 β†’ (πΉβ€˜π‘₯) = 𝑍) β†’ (𝐹 supp 𝑍) βŠ† (𝐺 supp 𝑍)))
 
Theoremfunsssuppss 8186 The support of a function which is a subset of another function is a subset of the support of this other function. (Contributed by AV, 27-Jul-2019.)
((Fun 𝐺 ∧ 𝐹 βŠ† 𝐺 ∧ 𝐺 ∈ 𝑉) β†’ (𝐹 supp 𝑍) βŠ† (𝐺 supp 𝑍))
 
Theoremfnsuppres 8187 Two ways to express restriction of a support set. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 28-May-2019.)
((𝐹 Fn (𝐴 βˆͺ 𝐡) ∧ (𝐹 ∈ π‘Š ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐡) = βˆ…) β†’ ((𝐹 supp 𝑍) βŠ† 𝐴 ↔ (𝐹 β†Ύ 𝐡) = (𝐡 Γ— {𝑍})))
 
Theoremfnsuppeq0 8188 The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 28-May-2019.)
((𝐹 Fn 𝐴 ∧ 𝐴 ∈ π‘Š ∧ 𝑍 ∈ 𝑉) β†’ ((𝐹 supp 𝑍) = βˆ… ↔ 𝐹 = (𝐴 Γ— {𝑍})))
 
Theoremfczsupp0 8189 The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)
((𝐡 Γ— {𝑍}) supp 𝑍) = βˆ…
 
Theoremsuppss 8190* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 5-Aug-2024.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ (𝐴 βˆ– π‘Š)) β†’ (πΉβ€˜π‘˜) = 𝑍)    β‡’   (πœ‘ β†’ (𝐹 supp 𝑍) βŠ† π‘Š)
 
TheoremsuppssOLD 8191* Obsolete version of suppss 8190 as of 5-Aug-2024. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   ((πœ‘ ∧ π‘˜ ∈ (𝐴 βˆ– π‘Š)) β†’ (πΉβ€˜π‘˜) = 𝑍)    β‡’   (πœ‘ β†’ (𝐹 supp 𝑍) βŠ† π‘Š)
 
Theoremsuppssr 8192 A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ (𝐹 supp 𝑍) βŠ† π‘Š)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 ∈ π‘ˆ)    β‡’   ((πœ‘ ∧ 𝑋 ∈ (𝐴 βˆ– π‘Š)) β†’ (πΉβ€˜π‘‹) = 𝑍)
 
Theoremsuppssrg 8193 A function is zero outside its support. Version of suppssr 8192 avoiding ax-rep 5279 by assuming 𝐹 is a set rather than its domain 𝐴. (Contributed by SN, 5-May-2024.)
(πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ (𝐹 supp 𝑍) βŠ† π‘Š)    &   (πœ‘ β†’ 𝐹 ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 ∈ π‘ˆ)    β‡’   ((πœ‘ ∧ 𝑋 ∈ (𝐴 βˆ– π‘Š)) β†’ (πΉβ€˜π‘‹) = 𝑍)
 
Theoremsuppssov1 8194* Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.) (Proof shortened by SN, 11-Apr-2025.)
(πœ‘ β†’ ((π‘₯ ∈ 𝐷 ↦ 𝐴) supp π‘Œ) βŠ† 𝐿)    &   ((πœ‘ ∧ 𝑣 ∈ 𝑅) β†’ (π‘Œπ‘‚π‘£) = 𝑍)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐷) β†’ 𝐴 ∈ 𝑉)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐷) β†’ 𝐡 ∈ 𝑅)    &   (πœ‘ β†’ π‘Œ ∈ π‘Š)    β‡’   (πœ‘ β†’ ((π‘₯ ∈ 𝐷 ↦ (𝐴𝑂𝐡)) supp 𝑍) βŠ† 𝐿)
 
Theoremsuppssov2 8195* Formula building theorem for support restrictions: operator with right annihilator. (Contributed by SN, 11-Apr-2025.)
(πœ‘ β†’ ((π‘₯ ∈ 𝐷 ↦ 𝐡) supp π‘Œ) βŠ† 𝐿)    &   ((πœ‘ ∧ 𝑣 ∈ 𝑅) β†’ (π‘£π‘‚π‘Œ) = 𝑍)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐷) β†’ 𝐴 ∈ 𝑅)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐷) β†’ 𝐡 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ π‘Š)    β‡’   (πœ‘ β†’ ((π‘₯ ∈ 𝐷 ↦ (𝐴𝑂𝐡)) supp 𝑍) βŠ† 𝐿)
 
Theoremsuppssof1 8196* Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
(πœ‘ β†’ (𝐴 supp π‘Œ) βŠ† 𝐿)    &   ((πœ‘ ∧ 𝑣 ∈ 𝑅) β†’ (π‘Œπ‘‚π‘£) = 𝑍)    &   (πœ‘ β†’ 𝐴:π·βŸΆπ‘‰)    &   (πœ‘ β†’ 𝐡:π·βŸΆπ‘…)    &   (πœ‘ β†’ 𝐷 ∈ π‘Š)    &   (πœ‘ β†’ π‘Œ ∈ π‘ˆ)    β‡’   (πœ‘ β†’ ((𝐴 ∘f 𝑂𝐡) supp 𝑍) βŠ† 𝐿)
 
Theoremsuppss2 8197* Show that the support of a function is contained in a set. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 22-Mar-2015.) (Revised by AV, 28-May-2019.)
((πœ‘ ∧ π‘˜ ∈ (𝐴 βˆ– π‘Š)) β†’ 𝐡 = 𝑍)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡) supp 𝑍) βŠ† π‘Š)
 
Theoremsuppsssn 8198* Show that the support of a function is a subset of a singleton. (Contributed by AV, 21-Jul-2019.)
((πœ‘ ∧ π‘˜ ∈ 𝐴 ∧ π‘˜ β‰  π‘Š) β†’ 𝐡 = 𝑍)    &   (πœ‘ β†’ 𝐴 ∈ 𝑉)    β‡’   (πœ‘ β†’ ((π‘˜ ∈ 𝐴 ↦ 𝐡) supp 𝑍) βŠ† {π‘Š})
 
Theoremsuppssfv 8199* Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 28-May-2019.)
(πœ‘ β†’ ((π‘₯ ∈ 𝐷 ↦ 𝐴) supp π‘Œ) βŠ† 𝐿)    &   (πœ‘ β†’ (πΉβ€˜π‘Œ) = 𝑍)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐷) β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ π‘ˆ)    β‡’   (πœ‘ β†’ ((π‘₯ ∈ 𝐷 ↦ (πΉβ€˜π΄)) supp 𝑍) βŠ† 𝐿)
 
Theoremsuppofssd 8200 Condition for the support of a function operation to be a subset of the union of the supports of the left and right function terms. (Contributed by Steven Nguyen, 28-Aug-2023.)
(πœ‘ β†’ 𝐴 ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 ∈ 𝐡)    &   (πœ‘ β†’ 𝐹:𝐴⟢𝐡)    &   (πœ‘ β†’ 𝐺:𝐴⟢𝐡)    &   (πœ‘ β†’ (𝑍𝑋𝑍) = 𝑍)    β‡’   (πœ‘ β†’ ((𝐹 ∘f 𝑋𝐺) supp 𝑍) βŠ† ((𝐹 supp 𝑍) βˆͺ (𝐺 supp 𝑍)))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48151
  Copyright terms: Public domain < Previous  Next >