P. 82 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 8101-8200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	offsplitfpar 8101	Express the function operation map ∘f by the functions defined in fsplit 8099 and fpar 8098. (Contributed by AV, 4-Jan-2024.)
⊢ 𝐻 = ((◡(1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   ⊢ 𝑆 = (◡(1st ↾ I ) ↾ 𝐴)    ⇒   ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) ∧ ( + Fn 𝐶 ∧ (ran 𝐹 × ran 𝐺) ⊆ 𝐶)) → ( + ∘ (𝐻 ∘ 𝑆)) = (𝐹 ∘f + 𝐺))
Theorem	f2ndf 8102	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⟶𝐵 → (2nd ↾ 𝐹):𝐹⟶𝐵)
Theorem	fo2ndf 8103	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⟶𝐵 → (2nd ↾ 𝐹):𝐹–onto→ran 𝐹)
Theorem	f1o2ndf1 8104	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→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹)
Theorem	opco1 8105	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴(𝐹 ∘ 1st )𝐵) = (𝐹‘𝐴))
Theorem	opco2 8106	Value of an operation precomposed with the projection on the second component. (Contributed by BJ, 27-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐴(𝐹 ∘ 2nd )𝐵) = (𝐹‘𝐵))
Theorem	opco1i 8107	Inference form of opco1 8105. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐵(𝐹 ∘ 1st )𝐶) = (𝐹‘𝐵)
Theorem	frxp 8108*	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 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ ((𝑅 Fr 𝐴 ∧ 𝑆 Fr 𝐵) → 𝑇 Fr (𝐴 × 𝐵))
Theorem	xporderlem 8109*	Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ (((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑))))
Theorem	poxp 8110*	A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ ((𝑅 Po 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐴 × 𝐵))
Theorem	soxp 8111*	A lexicographical ordering of two strictly ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ ((𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or (𝐴 × 𝐵))
Theorem	wexp 8112*	A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ ((1st ‘𝑥) = (1st ‘𝑦) ∧ (2nd ‘𝑥)𝑆(2nd ‘𝑦))))}    ⇒   ⊢ ((𝑅 We 𝐴 ∧ 𝑆 We 𝐵) → 𝑇 We (𝐴 × 𝐵))
Theorem	fnwelem 8113*	Lemma for fnwe 8114. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐵)    &   ⊢ (𝜑 → 𝑆 We 𝐴)    &   ⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V)    &   ⊢ 𝑄 = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝐵 × 𝐴) ∧ 𝑣 ∈ (𝐵 × 𝐴)) ∧ ((1st ‘𝑢)𝑅(1st ‘𝑣) ∨ ((1st ‘𝑢) = (1st ‘𝑣) ∧ (2nd ‘𝑢)𝑆(2nd ‘𝑣))))}    &   ⊢ 𝐺 = (𝑧 ∈ 𝐴 ↦ ⟨(𝐹‘𝑧), 𝑧⟩)    ⇒   ⊢ (𝜑 → 𝑇 We 𝐴)
Theorem	fnwe 8114*	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 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 We 𝐵)    &   ⊢ (𝜑 → 𝑆 We 𝐴)    &   ⊢ (𝜑 → (𝐹 “ 𝑤) ∈ V)    ⇒   ⊢ (𝜑 → 𝑇 We 𝐴)
Theorem	fnse 8115*	Condition for the well-order in fnwe 8114 to be set-like. (Contributed by Mario Carneiro, 25-Jun-2015.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))}    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝑅 Se 𝐵)    &   ⊢ (𝜑 → (◡𝐹 “ 𝑤) ∈ V)    ⇒   ⊢ (𝜑 → 𝑇 Se 𝐴)
Theorem	fvproj 8116*	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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻‘⟨𝑋, 𝑌⟩) = ⟨(𝐹‘𝑋), (𝐺‘𝑌)⟩)
Theorem	fimaproj 8117*	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 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑦)⟩)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻 “ (𝑋 × 𝑌)) = ((𝐹 “ 𝑋) × (𝐺 “ 𝑌)))
Theorem	ralxpes 8118*	A version of ralxp 5808 with explicit substitution. (Contributed by Scott Fenton, 21-Aug-2024.)
⊢ (∀𝑥 ∈ (𝐴 × 𝐵)[(1st ‘𝑥) / 𝑦][(2nd ‘𝑥) / 𝑧]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜑)
Theorem	ralxp3f 8119*	Restricted for all over a triple Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑤𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
Theorem	ralxp3 8120*	Restricted for all over a triple Cartesian product. (Contributed by Scott Fenton, 2-Feb-2025.)
⊢ (𝑥 = ⟨𝑦, 𝑧, 𝑤⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜓)
Theorem	ralxp3es 8121*	Restricted for-all over a triple Cartesian product with explicit substitution. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ (∀𝑥 ∈ ((𝐴 × 𝐵) × 𝐶)[(1st ‘(1st ‘𝑥)) / 𝑦][(2nd ‘(1st ‘𝑥)) / 𝑧][(2nd ‘𝑥) / 𝑤]𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝜑)
2.4.9  Induction on Cartesian products
Theorem	frpoins3xpg 8122*	Special case of founded partial induction over a Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (∀𝑧∀𝑤(⟨𝑧, 𝑤⟩ ∈ Pred(𝑅, (𝐴 × 𝐵), ⟨𝑥, 𝑦⟩) → 𝜒) → 𝜑))    &   ⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑤 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑦 = 𝑌 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (((𝑅 Fr (𝐴 × 𝐵) ∧ 𝑅 Po (𝐴 × 𝐵) ∧ 𝑅 Se (𝐴 × 𝐵)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝜏)
Theorem	frpoins3xp3g 8123*	Special case of founded partial recursion over a triple Cartesian product. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) → (∀𝑤∀𝑡∀𝑢(⟨𝑤, 𝑡, 𝑢⟩ ∈ Pred(𝑅, ((𝐴 × 𝐵) × 𝐶), ⟨𝑥, 𝑦, 𝑧⟩) → 𝜃) → 𝜑))    &   ⊢ (𝑥 = 𝑤 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝑡 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑧 = 𝑢 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑦 = 𝑌 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑧 = 𝑍 → (𝜂 ↔ 𝜁))    ⇒   ⊢ (((𝑅 Fr ((𝐴 × 𝐵) × 𝐶) ∧ 𝑅 Po ((𝐴 × 𝐵) × 𝐶) ∧ 𝑅 Se ((𝐴 × 𝐵) × 𝐶)) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶)) → 𝜁)
2.4.10  Ordering on Cartesian products
Theorem	xpord2lem 8124*	Lemma for Cartesian product ordering. Calculate the value of the Cartesian product relation. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐵) ∧ ((𝑎𝑅𝑐 ∨ 𝑎 = 𝑐) ∧ (𝑏𝑆𝑑 ∨ 𝑏 = 𝑑) ∧ (𝑎 ≠ 𝑐 ∨ 𝑏 ≠ 𝑑))))
Theorem	poxp2 8125*	Another way of partially ordering a Cartesian product of two classes. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Po 𝐴)    &   ⊢ (𝜑 → 𝑆 Po 𝐵)    ⇒   ⊢ (𝜑 → 𝑇 Po (𝐴 × 𝐵))
Theorem	frxp2 8126*	Another way of giving a well-founded order to a Cartesian product of two classes. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Fr 𝐴)    &   ⊢ (𝜑 → 𝑆 Fr 𝐵)    ⇒   ⊢ (𝜑 → 𝑇 Fr (𝐴 × 𝐵))
Theorem	xpord2pred 8127*	Calculate the predecessor class in frxp2 8126. (Contributed by Scott Fenton, 22-Aug-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → Pred(𝑇, (𝐴 × 𝐵), ⟨𝑋, 𝑌⟩) = (((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) ∖ {⟨𝑋, 𝑌⟩}))
Theorem	sexp2 8128*	Condition for the relation in frxp2 8126 to be set-like. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑆 Se 𝐵)    ⇒   ⊢ (𝜑 → 𝑇 Se (𝐴 × 𝐵))
Theorem	xpord2indlem 8129*	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 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵) ∧ (((1st ‘𝑥)𝑅(1st ‘𝑦) ∨ (1st ‘𝑥) = (1st ‘𝑦)) ∧ ((2nd ‘𝑥)𝑆(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦)) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ 𝑅 Fr 𝐴    &   ⊢ 𝑅 Po 𝐴    &   ⊢ 𝑅 Se 𝐴    &   ⊢ 𝑆 Fr 𝐵    &   ⊢ 𝑆 Po 𝐵    &   ⊢ 𝑆 Se 𝐵    &   ⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))    &   ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)
Theorem	xpord2ind 8130*	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    &   ⊢ 𝑅 Po 𝐴    &   ⊢ 𝑅 Se 𝐴    &   ⊢ 𝑆 Fr 𝐵    &   ⊢ 𝑆 Po 𝐵    &   ⊢ 𝑆 Se 𝐵    &   ⊢ (𝑎 = 𝑐 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑎 = 𝑐 → (𝜃 ↔ 𝜒))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑌 → (𝜏 ↔ 𝜂))    &   ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑐 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑑 ∈ Pred (𝑆, 𝐵, 𝑏)𝜃) → 𝜑))    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝜂)
Theorem	xpord3lem 8131*	Lemma for triple ordering. Calculate the value of the relation. (Contributed by Scott Fenton, 21-Aug-2024.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ (⟨𝑎, 𝑏, 𝑐⟩𝑈⟨𝑑, 𝑒, 𝑓⟩ ↔ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) ∧ (𝑑 ∈ 𝐴 ∧ 𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐶) ∧ (((𝑎𝑅𝑑 ∨ 𝑎 = 𝑑) ∧ (𝑏𝑆𝑒 ∨ 𝑏 = 𝑒) ∧ (𝑐𝑇𝑓 ∨ 𝑐 = 𝑓)) ∧ (𝑎 ≠ 𝑑 ∨ 𝑏 ≠ 𝑒 ∨ 𝑐 ≠ 𝑓))))
Theorem	poxp3 8132*	Triple Cartesian product partial ordering. (Contributed by Scott Fenton, 21-Aug-2024.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Po 𝐴)    &   ⊢ (𝜑 → 𝑆 Po 𝐵)    &   ⊢ (𝜑 → 𝑇 Po 𝐶)    ⇒   ⊢ (𝜑 → 𝑈 Po ((𝐴 × 𝐵) × 𝐶))
Theorem	frxp3 8133*	Give well-foundedness over a triple Cartesian product. (Contributed by Scott Fenton, 21-Aug-2024.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Fr 𝐴)    &   ⊢ (𝜑 → 𝑆 Fr 𝐵)    &   ⊢ (𝜑 → 𝑇 Fr 𝐶)    ⇒   ⊢ (𝜑 → 𝑈 Fr ((𝐴 × 𝐵) × 𝐶))
Theorem	xpord3pred 8134*	Calculate the predecsessor class for the triple order. (Contributed by Scott Fenton, 31-Jan-2025.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → Pred(𝑈, ((𝐴 × 𝐵) × 𝐶), ⟨𝑋, 𝑌, 𝑍⟩) = ((((Pred(𝑅, 𝐴, 𝑋) ∪ {𝑋}) × (Pred(𝑆, 𝐵, 𝑌) ∪ {𝑌})) × (Pred(𝑇, 𝐶, 𝑍) ∪ {𝑍})) ∖ {⟨𝑋, 𝑌, 𝑍⟩}))
Theorem	sexp3 8135*	Show that the triple order is set-like. (Contributed by Scott Fenton, 21-Aug-2024.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑆 Se 𝐵)    &   ⊢ (𝜑 → 𝑇 Se 𝐶)    ⇒   ⊢ (𝜑 → 𝑈 Se ((𝐴 × 𝐵) × 𝐶))
Theorem	xpord3inddlem 8136*	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.)
⊢ 𝑈 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ ((((1st ‘(1st ‘𝑥))𝑅(1st ‘(1st ‘𝑦)) ∨ (1st ‘(1st ‘𝑥)) = (1st ‘(1st ‘𝑦))) ∧ ((2nd ‘(1st ‘𝑥))𝑆(2nd ‘(1st ‘𝑦)) ∨ (2nd ‘(1st ‘𝑥)) = (2nd ‘(1st ‘𝑦))) ∧ ((2nd ‘𝑥)𝑇(2nd ‘𝑦) ∨ (2nd ‘𝑥) = (2nd ‘𝑦))) ∧ 𝑥 ≠ 𝑦))}    &   ⊢ (𝜅 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜅 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜅 → 𝑍 ∈ 𝐶)    &   ⊢ (𝜅 → 𝑅 Fr 𝐴)    &   ⊢ (𝜅 → 𝑅 Po 𝐴)    &   ⊢ (𝜅 → 𝑅 Se 𝐴)    &   ⊢ (𝜅 → 𝑆 Fr 𝐵)    &   ⊢ (𝜅 → 𝑆 Po 𝐵)    &   ⊢ (𝜅 → 𝑆 Se 𝐵)    &   ⊢ (𝜅 → 𝑇 Fr 𝐶)    &   ⊢ (𝜅 → 𝑇 Po 𝐶)    &   ⊢ (𝜅 → 𝑇 Se 𝐶)    &   ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ⊢ ((𝜅 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑))    ⇒   ⊢ (𝜅 → 𝜆)
Theorem	xpord3indd 8137*	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.)
⊢ (𝜅 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜅 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜅 → 𝑍 ∈ 𝐶)    &   ⊢ (𝜅 → 𝑅 Fr 𝐴)    &   ⊢ (𝜅 → 𝑅 Po 𝐴)    &   ⊢ (𝜅 → 𝑅 Se 𝐴)    &   ⊢ (𝜅 → 𝑆 Fr 𝐵)    &   ⊢ (𝜅 → 𝑆 Po 𝐵)    &   ⊢ (𝜅 → 𝑆 Se 𝐵)    &   ⊢ (𝜅 → 𝑇 Fr 𝐶)    &   ⊢ (𝜅 → 𝑇 Po 𝐶)    &   ⊢ (𝜅 → 𝑇 Se 𝐶)    &   ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ⊢ ((𝜅 ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶)) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑))    ⇒   ⊢ (𝜅 → 𝜆)
Theorem	xpord3ind 8138*	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    &   ⊢ 𝑅 Po 𝐴    &   ⊢ 𝑅 Se 𝐴    &   ⊢ 𝑆 Fr 𝐵    &   ⊢ 𝑆 Po 𝐵    &   ⊢ 𝑆 Se 𝐵    &   ⊢ 𝑇 Fr 𝐶    &   ⊢ 𝑇 Po 𝐶    &   ⊢ 𝑇 Se 𝐶    &   ⊢ (𝑎 = 𝑑 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝑒 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝑓 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑎 = 𝑑 → (𝜏 ↔ 𝜃))    &   ⊢ (𝑏 = 𝑒 → (𝜂 ↔ 𝜏))    &   ⊢ (𝑏 = 𝑒 → (𝜁 ↔ 𝜃))    &   ⊢ (𝑐 = 𝑓 → (𝜎 ↔ 𝜏))    &   ⊢ (𝑎 = 𝑋 → (𝜑 ↔ 𝜌))    &   ⊢ (𝑏 = 𝑌 → (𝜌 ↔ 𝜇))    &   ⊢ (𝑐 = 𝑍 → (𝜇 ↔ 𝜆))    &   ⊢ ((𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) → (((∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜃 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜒 ∧ ∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜁) ∧ (∀𝑑 ∈ Pred (𝑅, 𝐴, 𝑎)𝜓 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜏 ∧ ∀𝑒 ∈ Pred (𝑆, 𝐵, 𝑏)𝜎) ∧ ∀𝑓 ∈ Pred (𝑇, 𝐶, 𝑐)𝜂) → 𝜑))    ⇒   ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐶) → 𝜆)
2.4.11  Ordering Ordinal Sequences
Theorem	orderseqlem 8139*	Lemma for poseq 8140 and soseq 8141. The function value of a sequence is either in 𝐴 or null. (Contributed by Scott Fenton, 8-Jun-2011.)
⊢ 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶𝐴}    ⇒   ⊢ (𝐺 ∈ 𝐹 → (𝐺‘𝑋) ∈ (𝐴 ∪ {∅}))
Theorem	poseq 8140*	A partial ordering of ordinal sequences. (Contributed by Scott Fenton, 8-Jun-2011.)
⊢ 𝑅 Po (𝐴 ∪ {∅})    &   ⊢ 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶𝐴}    &   ⊢ 𝑆 = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥)𝑅(𝑔‘𝑥)))}    ⇒   ⊢ 𝑆 Po 𝐹
Theorem	soseq 8141*	A linear ordering of ordinal sequences. (Contributed by Scott Fenton, 8-Jun-2011.)
⊢ 𝑅 Or (𝐴 ∪ {∅})    &   ⊢ 𝐹 = {𝑓 ∣ ∃𝑥 ∈ On 𝑓:𝑥⟶𝐴}    &   ⊢ 𝑆 = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ 𝐹 ∧ 𝑔 ∈ 𝐹) ∧ ∃𝑥 ∈ On (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑔‘𝑦) ∧ (𝑓‘𝑥)𝑅(𝑔‘𝑥)))}    &   ⊢ ¬ ∅ ∈ 𝐴    ⇒   ⊢ 𝑆 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 8144) are based on the Axiom of Union (usage of dmexg 7880), these definition and theorems cannot be provided earlier. Until April 2019, the support of a function was represented by the expression (◡𝑅 “ (V ∖ {𝑍})) (see suppimacnv 8156). 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).
Syntax	csupp 8142	Extend class definition to include the support of functions.
class supp
Definition	df-supp 8143*	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 = (𝑥 ∈ V, 𝑧 ∈ V ↦ {𝑖 ∈ dom 𝑥 ∣ (𝑥 “ {𝑖}) ≠ {𝑧}})
Theorem	suppval 8144*	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 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋 “ {𝑖}) ≠ {𝑍}})
Theorem	supp0prc 8145	The support of a class is empty if either the class or the "zero" is a proper class. (Contributed by AV, 28-May-2019.)
⊢ (¬ (𝑋 ∈ V ∧ 𝑍 ∈ V) → (𝑋 supp 𝑍) = ∅)
Theorem	suppvalbr 8146*	The value of the operation constructing the support of a function expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))})
Theorem	supp0 8147	The support of the empty set is the empty set. (Contributed by AV, 12-Apr-2019.)
⊢ (𝑍 ∈ 𝑊 → (∅ supp 𝑍) = ∅)
Theorem	suppval1 8148*	The value of the operation constructing the support of a function. (Contributed by AV, 6-Apr-2019.)
⊢ ((Fun 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑋 supp 𝑍) = {𝑖 ∈ dom 𝑋 ∣ (𝑋‘𝑖) ≠ 𝑍})
Theorem	suppvalfng 8149*	The value of the operation constructing the support of a function with a given domain. This version of suppvalfn 8150 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5237. (Contributed by SN, 5-Aug-2024.)
⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})
Theorem	suppvalfn 8150*	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 ∈ 𝑊) → (𝐹 supp 𝑍) = {𝑖 ∈ 𝑋 ∣ (𝐹‘𝑖) ≠ 𝑍})
Theorem	elsuppfng 8151	An element of the support of a function with a given domain. This version of elsuppfn 8152 assumes 𝐹 is a set rather than its domain 𝑋, avoiding ax-rep 5237. (Contributed by SN, 5-Aug-2024.)
⊢ ((𝐹 Fn 𝑋 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆 ∈ 𝑋 ∧ (𝐹‘𝑆) ≠ 𝑍)))
Theorem	elsuppfn 8152	An element of the support of a function with a given domain. (Contributed by AV, 27-May-2019.)
⊢ ((𝐹 Fn 𝑋 ∧ 𝑋 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑆 ∈ (𝐹 supp 𝑍) ↔ (𝑆 ∈ 𝑋 ∧ (𝐹‘𝑆) ≠ 𝑍)))
Theorem	fvdifsupp 8153	Function value is zero outside of its support. (Contributed by Thierry Arnoux, 21-Jan-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∖ (𝐹 supp 𝑍)))    ⇒   ⊢ (𝜑 → (𝐹‘𝑋) = 𝑍)
Theorem	cnvimadfsn 8154*	The support of functions "defined" by inverse images expressed by binary relations. (Contributed by AV, 7-Apr-2019.)
⊢ (◡𝑅 “ (V ∖ {𝑍})) = {𝑥 ∣ ∃𝑦(𝑥𝑅𝑦 ∧ 𝑦 ≠ 𝑍)}
Theorem	suppimacnvss 8155	The support of functions "defined" by inverse images is a subset of the support defined by df-supp 8143. (Contributed by AV, 7-Apr-2019.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (◡𝑅 “ (V ∖ {𝑍})) ⊆ (𝑅 supp 𝑍))
Theorem	suppimacnv 8156	Support sets of functions expressed by inverse images. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 7-Apr-2019.)
⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = (◡𝑅 “ (V ∖ {𝑍})))
Theorem	fsuppeq 8157	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 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))))
Theorem	fsuppeqg 8158	Version of fsuppeq 8157 avoiding ax-rep 5237 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 30-Jul-2024.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹:𝐼⟶𝑆 → (𝐹 supp 𝑍) = (◡𝐹 “ (𝑆 ∖ {𝑍}))))
Theorem	suppssdm 8159	The support of a function is a subset of the function's domain. (Contributed by AV, 30-May-2019.)
⊢ (𝐹 supp 𝑍) ⊆ dom 𝐹
Theorem	suppsnop 8160	The support of a singleton of an ordered pair. (Contributed by AV, 12-Apr-2019.)
⊢ 𝐹 = {⟨𝑋, 𝑌⟩}    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑈) → (𝐹 supp 𝑍) = if(𝑌 = 𝑍, ∅, {𝑋}))
Theorem	snopsuppss 8161	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.)
⊢ ({⟨𝑋, 𝑌⟩} supp 𝑍) ⊆ {𝑋}
Theorem	fvn0elsupp 8162	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.)
⊢ (((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵) ∧ (𝐺 Fn 𝐵 ∧ (𝐺‘𝑋) ≠ ∅)) → 𝑋 ∈ (𝐺 supp ∅))
Theorem	fvn0elsuppb 8163	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.)
⊢ ((𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝐵 ∧ 𝐺 Fn 𝐵) → ((𝐺‘𝑋) ≠ ∅ ↔ 𝑋 ∈ (𝐺 supp ∅)))
Theorem	rexsupp 8164*	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 ∈ 𝑊) → (∃𝑥 ∈ (𝐹 supp 𝑍)𝜑 ↔ ∃𝑥 ∈ 𝑋 ((𝐹‘𝑥) ≠ 𝑍 ∧ 𝜑)))
Theorem	ressuppss 8165	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 ∈ 𝑊) → ((𝐹 ↾ 𝐵) supp 𝑍) ⊆ (𝐹 supp 𝑍))
Theorem	suppun 8166	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.)
⊢ (𝜑 → 𝐺 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹 ∪ 𝐺) supp 𝑍))
Theorem	ressuppssdif 8167	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 ∈ 𝑊) → (𝐹 supp 𝑍) ⊆ (((𝐹 ↾ 𝐵) supp 𝑍) ∪ (dom 𝐹 ∖ 𝐵)))
Theorem	mptsuppdifd 8168*	The support of a function in maps-to notation with a class difference. (Contributed by AV, 28-May-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ (V ∖ {𝑍})})
Theorem	mptsuppd 8169*	The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑈)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) = {𝑥 ∈ 𝐴 ∣ 𝐵 ≠ 𝑍})
Theorem	extmptsuppeq 8170*	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.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐵 ∖ 𝐴)) → 𝑋 = 𝑍)    ⇒   ⊢ (𝜑 → ((𝑛 ∈ 𝐴 ↦ 𝑋) supp 𝑍) = ((𝑛 ∈ 𝐵 ↦ 𝑋) supp 𝑍))
Theorem	suppfnss 8171*	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 𝐵) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊)) → (∀𝑥 ∈ 𝐴 ((𝐺‘𝑥) = 𝑍 → (𝐹‘𝑥) = 𝑍) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍)))
Theorem	funsssuppss 8172	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 ⊆ 𝐺 ∧ 𝐺 ∈ 𝑉) → (𝐹 supp 𝑍) ⊆ (𝐺 supp 𝑍))
Theorem	fnsuppres 8173	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 ∪ 𝐵) ∧ (𝐹 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 supp 𝑍) ⊆ 𝐴 ↔ (𝐹 ↾ 𝐵) = (𝐵 × {𝑍})))
Theorem	fnsuppeq0 8174	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 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉) → ((𝐹 supp 𝑍) = ∅ ↔ 𝐹 = (𝐴 × {𝑍})))
Theorem	fczsupp0 8175	The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)
⊢ ((𝐵 × {𝑍}) supp 𝑍) = ∅
Theorem	suppss 8176*	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.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑘) = 𝑍)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)
Theorem	suppssr 8177	A function is zero outside its support. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by AV, 28-May-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)
Theorem	suppssrg 8178	A function is zero outside its support. Version of suppssr 8177 avoiding ax-rep 5237 by assuming 𝐹 is a set rather than its domain 𝐴. (Contributed by SN, 5-May-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐴 ∖ 𝑊)) → (𝐹‘𝑋) = 𝑍)
Theorem	suppssov1 8179*	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.)
⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ 𝐿)    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑅)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
Theorem	suppssov2 8180*	Formula building theorem for support restrictions: operator with right annihilator. (Contributed by SN, 11-Apr-2025.)
⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵) supp 𝑌) ⊆ 𝐿)    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑣𝑂𝑌) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐴𝑂𝐵)) supp 𝑍) ⊆ 𝐿)
Theorem	suppssof1 8181*	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.)
⊢ (𝜑 → (𝐴 supp 𝑌) ⊆ 𝐿)    &   ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑅) → (𝑌𝑂𝑣) = 𝑍)    &   ⊢ (𝜑 → 𝐴:𝐷⟶𝑉)    &   ⊢ (𝜑 → 𝐵:𝐷⟶𝑅)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝐴 ∘f 𝑂𝐵) supp 𝑍) ⊆ 𝐿)
Theorem	suppss2 8182*	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.)
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊)
Theorem	suppsssn 8183*	Show that the support of a function is a subset of a singleton. (Contributed by AV, 21-Jul-2019.)
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴 ∧ 𝑘 ≠ 𝑊) → 𝐵 = 𝑍)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ {𝑊})
Theorem	suppssfv 8184*	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.)
⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐴) supp 𝑌) ⊆ 𝐿)    &   ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    ⇒   ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ (𝐹‘𝐴)) supp 𝑍) ⊆ 𝐿)
Theorem	suppofssd 8185	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.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝑍𝑋𝑍) = 𝑍)    ⇒   ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))
Theorem	suppofss1d 8186*	Condition for the support of a function operation to be a subset of the support of the left function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑍𝑋𝑥) = 𝑍)    ⇒   ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑍))
Theorem	suppofss2d 8187*	Condition for the support of a function operation to be a subset of the support of the right function term. (Contributed by Thierry Arnoux, 21-Jun-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥𝑋𝑍) = 𝑍)    ⇒   ⊢ (𝜑 → ((𝐹 ∘f 𝑋𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑍))
Theorem	suppco 8188	The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) Extract this statement from the proof of supp0cosupp0 8190. (Revised by SN, 15-Sep-2023.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘ 𝐺) supp 𝑍) = (◡𝐺 “ (𝐹 supp 𝑍)))
Theorem	suppcoss 8189	The support of the composition of two functions is a subset of the support of the inner function if the outer function preserves zero. Compare suppssfv 8184, which has a sethood condition on 𝐴 instead of 𝐵. (Contributed by SN, 25-May-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐹‘𝑌) = 𝑍)    ⇒   ⊢ (𝜑 → ((𝐹 ∘ 𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))
Theorem	supp0cosupp0 8190	The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹 ∘ 𝐺) supp 𝑍) = ∅))
Theorem	imacosupp 8191	The image of the support of the composition of two functions is the support of the outer function. (Contributed by AV, 30-May-2019.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((Fun 𝐺 ∧ (𝐹 supp 𝑍) ⊆ ran 𝐺) → (𝐺 “ ((𝐹 ∘ 𝐺) supp 𝑍)) = (𝐹 supp 𝑍)))
2.4.13  Special maps-to operations The following theorems are about maps-to operations (see df-mpo 7395) where the domain of the second argument depends on the domain of the first argument, especially when the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpox" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpox 7485, ovmpox 7545 and fmpox 8049). If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpoxop", and the maps-to operations are called "x-op maps-to operations" for short.
Theorem	opeliunxp2f 8192*	Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 5805. (Contributed by AV, 25-Oct-2020.)
⊢ Ⅎ𝑥𝐸    &   ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸)    ⇒   ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))
Theorem	mpoxeldm 8193*	If there is an element of the value of an operation given by a maps-to rule, then the first argument is an element of the first component of the domain and the second argument is an element of the second component of the domain depending on the first argument. (Contributed by AV, 25-Oct-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    ⇒   ⊢ (𝑁 ∈ (𝑋𝐹𝑌) → (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ ⦋𝑋 / 𝑥⦌𝐷))
Theorem	mpoxneldm 8194*	If the first argument of an operation given by a maps-to rule is not an element of the first component of the domain or the second argument is not an element of the second component of the domain depending on the first argument, then the value of the operation is the empty set. (Contributed by AV, 25-Oct-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)    ⇒   ⊢ ((𝑋 ∉ 𝐶 ∨ 𝑌 ∉ ⦋𝑋 / 𝑥⦌𝐷) → (𝑋𝐹𝑌) = ∅)
Theorem	mpoxopn0yelv 8195*	If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉))
Theorem	mpoxopynvov0g 8196*	If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∉ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
Theorem	mpoxopxnop0 8197*	If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is not an ordered pair, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ (¬ 𝑉 ∈ (V × V) → (𝑉𝐹𝐾) = ∅)
Theorem	mpoxopx0ov0 8198*	If the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, is the empty set, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ (∅𝐹𝐾) = ∅
Theorem	mpoxopxprcov0 8199*	If the components of the first argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, are not sets, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ (¬ (𝑉 ∈ V ∧ 𝑊 ∈ V) → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)
Theorem	mpoxopynvov0 8200*	If the second argument of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument is not element of the first component of the first argument, then the value of the operation is the empty set. (Contributed by Alexander van der Vekens, 10-Oct-2017.)
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶)    ⇒   ⊢ (𝐾 ∉ 𝑉 → (⟨𝑉, 𝑊⟩𝐹𝐾) = ∅)