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Theorem List for Metamath Proof Explorer - 30701-30800   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.8.4  Misc. Useful Theorems

Theoremnepss 30701 Two classes are inequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)
(𝐴𝐵 ↔ ((𝐴𝐵) ⊊ 𝐴 ∨ (𝐴𝐵) ⊊ 𝐵))

Theorem3ccased 30702 Triple disjunction form of ccased 984. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝜑 → ((𝜒𝜂) → 𝜓))    &   (𝜑 → ((𝜒𝜁) → 𝜓))    &   (𝜑 → ((𝜒𝜎) → 𝜓))    &   (𝜑 → ((𝜃𝜂) → 𝜓))    &   (𝜑 → ((𝜃𝜁) → 𝜓))    &   (𝜑 → ((𝜃𝜎) → 𝜓))    &   (𝜑 → ((𝜏𝜂) → 𝜓))    &   (𝜑 → ((𝜏𝜁) → 𝜓))    &   (𝜑 → ((𝜏𝜎) → 𝜓))       (𝜑 → (((𝜒𝜃𝜏) ∧ (𝜂𝜁𝜎)) → 𝜓))

Theoremdfso3 30703* Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)
(𝑅 Or 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))

Theorembrtpid1 30704 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵

Theorembrtpid2 30705 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵

Theorembrtpid3 30706 A binary relationship involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.)
𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵

Theoremceqsrexv2 30707* Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))

Theoremiota5f 30708* A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)
𝑥𝜑    &   𝑥𝐴    &   ((𝜑𝐴𝑉) → (𝜓𝑥 = 𝐴))       ((𝜑𝐴𝑉) → (℩𝑥𝜓) = 𝐴)

Theoremceqsralv2 30709* Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.)
(𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 (𝑥 = 𝐴𝜑) ↔ (𝐴𝐵𝜓))

20.8.5  Properties of real and complex numbers

Theoremsqdivzi 30710 Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.)
𝐴 ∈ ℂ    &   𝐵 ∈ ℂ       (𝐵 ≠ 0 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))

Theoremsubdivcomb1 30711 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → (((𝐶 · 𝐴) − 𝐵) / 𝐶) = (𝐴 − (𝐵 / 𝐶)))

Theoremsubdivcomb2 30712 Bring a term in a subtraction into the numerator. (Contributed by Scott Fenton, 3-Jul-2013.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 ≠ 0)) → ((𝐴 − (𝐶 · 𝐵)) / 𝐶) = ((𝐴 / 𝐶) − 𝐵))

Theoremsupfz 30713 The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
(𝑁 ∈ (ℤ𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁)

Theoreminffz 30714 The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
(𝑁 ∈ (ℤ𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑀)

Theoremfz0n 30715 The sequence (0...(𝑁 − 1)) is empty iff 𝑁 is zero. (Contributed by Scott Fenton, 16-May-2014.)
(𝑁 ∈ ℕ0 → ((0...(𝑁 − 1)) = ∅ ↔ 𝑁 = 0))

Theoremshftvalg 30716 Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.)
((𝐹𝑉𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵𝐴)))

Theoremdivcnvlin 30717* Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℤ)    &   (𝜑𝐹𝑉)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = ((𝑘 + 𝐴) / (𝑘 + 𝐵)))       (𝜑𝐹 ⇝ 1)

Theoremclimlec3 30718* Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐹𝐴)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) ≤ 𝐵)       (𝜑𝐴𝐵)

Theoremlogi 30719 Calculate the logarithm of i. (Contributed by Scott Fenton, 13-Apr-2020.)
(log‘i) = (i · (π / 2))

Theoremiexpire 30720 i raised to itself is real. (Contributed by Scott Fenton, 13-Apr-2020.)
(i↑𝑐i) ∈ ℝ

Theorembcneg1 30721 The binomial coefficent over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.)
(𝑁 ∈ ℕ0 → (𝑁C-1) = 0)

Theorembcm1nt 30722 The proportion of one bionmial coefficient to another with 𝑁 decreased by 1. (Contributed by Scott Fenton, 23-Jun-2020.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (𝑁C𝐾) = (((𝑁 − 1)C𝐾) · (𝑁 / (𝑁𝐾))))

Theorembcprod 30723* A product identity for binomial coefficents. (Contributed by Scott Fenton, 23-Jun-2020.)
(𝑁 ∈ ℕ → ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁)))

Theorembccolsum 30724* A column-sum rule for binomial coefficents. (Contributed by Scott Fenton, 24-Jun-2020.)
((𝑁 ∈ ℕ0𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1)))

20.8.6  Infinite products

Theoremiprodefisumlem 30725 Lemma for iprodefisum 30726. (Contributed by Scott Fenton, 11-Feb-2018.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹:𝑍⟶ℂ)       (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))

Theoremiprodefisum 30726* Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐵)    &   ((𝜑𝑘𝑍) → 𝐵 ∈ ℂ)    &   (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )       (𝜑 → ∏𝑘𝑍 (exp‘𝐵) = (exp‘Σ𝑘𝑍 𝐵))

Theoremiprodgam 30727* An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.)
(𝜑𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))       (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴))

20.8.7  Factorial limits

Theoremfaclimlem1 30728* Lemma for faclim 30731. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.)
(𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) = (𝑥 ∈ ℕ ↦ ((𝑀 + 1) · ((𝑥 + 1) / (𝑥 + (𝑀 + 1))))))

Theoremfaclimlem2 30729* Lemma for faclim 30731. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.)
(𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) ⇝ (𝑀 + 1))

Theoremfaclimlem3 30730 Lemma for faclim 30731. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.)
((𝑀 ∈ ℕ0𝐵 ∈ ℕ) → (((1 + (1 / 𝐵))↑(𝑀 + 1)) / (1 + ((𝑀 + 1) / 𝐵))) = ((((1 + (1 / 𝐵))↑𝑀) / (1 + (𝑀 / 𝐵))) · (((1 + (𝑀 / 𝐵)) · (1 + (1 / 𝐵))) / (1 + ((𝑀 + 1) / 𝐵)))))

Theoremfaclim 30731* An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)
𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))       (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴))

Theoremiprodfac 30732* An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.)
(𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘))))

Theoremfaclim2 30733* Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.)
𝐹 = (𝑛 ∈ ℕ ↦ (((!‘𝑛) · ((𝑛 + 1)↑𝑀)) / (!‘(𝑛 + 𝑀))))       (𝑀 ∈ ℕ0𝐹 ⇝ 1)

20.8.8  Greatest common divisor and divisibility

Theorempdivsq 30734 Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ) → (𝑃𝑀𝑃 ∥ (𝑀↑2)))

Theoremdvdspw 30735 Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐾𝑀𝐾 ∥ (𝑀𝑁)))

Theoremgcd32 30736 Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = ((𝐴 gcd 𝐶) gcd 𝐵))

Theoremgcdabsorb 30737 Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐵) = (𝐴 gcd 𝐵))

20.8.9  Properties of relationships

Theorembrtp 30738 A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)
𝑋 ∈ V    &   𝑌 ∈ V       (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}𝑌 ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷) ∨ (𝑋 = 𝐸𝑌 = 𝐹)))

Theoremdftr6 30739 A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
𝐴 ∈ V       (Tr 𝐴𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))

Theoremcoep 30740* Composition with epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥𝐵 𝐴𝑅𝑥)

Theoremcoepr 30741* Composition with the converse of epsilon. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴(𝑅 E )𝐵 ↔ ∃𝑥𝐴 𝑥𝑅𝐵)

Theoremdffr5 30742 A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
(𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ 𝑅)))

Theoremdfso2 30743 Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
(𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ 𝑅))))

Theoremdfpo2 30744 Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.)
(𝑅 Po 𝐴 ↔ ((𝑅 ∩ ( I ↾ 𝐴)) = ∅ ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅))

Theorembr8 30745* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
(𝑎 = 𝐴 → (𝜑𝜓))    &   (𝑏 = 𝐵 → (𝜓𝜒))    &   (𝑐 = 𝐶 → (𝜒𝜃))    &   (𝑑 = 𝐷 → (𝜃𝜏))    &   (𝑒 = 𝐸 → (𝜏𝜂))    &   (𝑓 = 𝐹 → (𝜂𝜁))    &   (𝑔 = 𝐺 → (𝜁𝜎))    &   ( = 𝐻 → (𝜎𝜌))    &   (𝑥 = 𝑋𝑃 = 𝑄)    &   𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃𝑒𝑃𝑓𝑃𝑔𝑃𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ⟩⟩ ∧ 𝜑)}       (((𝑋𝑆𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄𝐸𝑄) ∧ (𝐹𝑄𝐺𝑄𝐻𝑄)) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜌))

Theorembr6 30746* Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
(𝑎 = 𝐴 → (𝜑𝜓))    &   (𝑏 = 𝐵 → (𝜓𝜒))    &   (𝑐 = 𝐶 → (𝜒𝜃))    &   (𝑑 = 𝐷 → (𝜃𝜏))    &   (𝑒 = 𝐸 → (𝜏𝜂))    &   (𝑓 = 𝐹 → (𝜂𝜁))    &   (𝑥 = 𝑋𝑃 = 𝑄)    &   𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃𝑒𝑃𝑓𝑃 (𝑝 = ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ ∧ 𝑞 = ⟨𝑑, ⟨𝑒, 𝑓⟩⟩ ∧ 𝜑)}       ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄𝐶𝑄) ∧ (𝐷𝑄𝐸𝑄𝐹𝑄)) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩𝑅𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ 𝜁))

Theorembr4 30747* Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
(𝑎 = 𝐴 → (𝜑𝜓))    &   (𝑏 = 𝐵 → (𝜓𝜒))    &   (𝑐 = 𝐶 → (𝜒𝜃))    &   (𝑑 = 𝐷 → (𝜃𝜏))    &   (𝑥 = 𝑋𝑃 = 𝑄)    &   𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥𝑆𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)}       ((𝑋𝑆 ∧ (𝐴𝑄𝐵𝑄) ∧ (𝐶𝑄𝐷𝑄)) → (⟨𝐴, 𝐵𝑅𝐶, 𝐷⟩ ↔ 𝜏))

Theoremdfres3 30748 Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐴𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴))

Theoremcnvco1 30749 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
(𝐴𝐵) = (𝐵𝐴)

Theoremcnvco2 30750 Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
(𝐴𝐵) = (𝐵𝐴)

Theoremeldm3 30751 Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)
(𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)

Theoremelrn3 30752 Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)
(𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)

Theorempocnv 30753 The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 Po 𝐴𝑅 Po 𝐴)

Theoremsocnv 30754 The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
(𝑅 Or 𝐴𝑅 Or 𝐴)

20.8.10  Properties of functions and mappings

Theoremfunpsstri 30755 A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
((Fun 𝐻 ∧ (𝐹𝐻𝐺𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹𝐺𝐹 = 𝐺𝐺𝐹))

Theoremfundmpss 30756 If a class 𝐹 is a proper subset of a function 𝐺, then dom 𝐹 ⊊ dom 𝐺. (Contributed by Scott Fenton, 20-Apr-2011.)
(Fun 𝐺 → (𝐹𝐺 → dom 𝐹 ⊊ dom 𝐺))

Theoremfvresval 30757 The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.)
(((𝐹𝐵)‘𝐴) = (𝐹𝐴) ∨ ((𝐹𝐵)‘𝐴) = ∅)

Theoremfunsseq 30758 Given two functions with equal domains, equality only requires one direction of the subset relationship. (Contributed by Scott Fenton, 24-Apr-2012.) (Proof shortened by Mario Carneiro, 3-May-2015.)
((Fun 𝐹 ∧ Fun 𝐺 ∧ dom 𝐹 = dom 𝐺) → (𝐹 = 𝐺𝐹𝐺))

Theoremfununiq 30759 The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (Fun 𝐹 → ((𝐴𝐹𝐵𝐴𝐹𝐶) → 𝐵 = 𝐶))

Theoremfunbreq 30760 An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       ((Fun 𝐹𝐴𝐹𝐵) → (𝐴𝐹𝐶𝐵 = 𝐶))

Theoremmpteq12d 30761 An equality inference for the maps to notation. Compare mpteq12dv 4561. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝜑    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremfprb 30762* A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐴𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))

Theorembr1steq 30763 Uniqueness condition for binary relationship over the 1st relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)

Theorembr2ndeq 30764 Uniqueness condition for binary relationship over the 2nd relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)

Theorembr1steqg 30765 Uniqueness condition for binary relationship over the 1st relationship. (Contributed by Scott Fenton, 2-Jul-2020.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))

Theorembr2ndeqg 30766 Uniqueness condition for binary relationship over the 2nd relationship. (Contributed by Scott Fenton, 2-Jul-2020.)
((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵))

Theoremdfdm5 30767 Definition of domain in terms of 1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)

Theoremdfrn5 30768 Definition of range in terms of 2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)

Theoremopelco3 30769 Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.)
(⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))

Theoremelima4 30770 Quantifier-free expression saying that a class is a member of an image. (Contributed by Scott Fenton, 8-May-2018.)
(𝐴 ∈ (𝑅𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅)

Theoremfv1stcnv 30771 The value of the converse of 1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
((𝑋𝐴𝑌𝑉) → ((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)

Theoremfv2ndcnv 30772 The value of the converse of 2nd restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
((𝑋𝑉𝑌𝐴) → ((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)

20.8.11  Epsilon induction

Theoremsetinds 30773* Principle of E induction (set induction). If a property passes from all elements of 𝑥 to 𝑥 itself, then it holds for all 𝑥. (Contributed by Scott Fenton, 10-Mar-2011.)
(∀𝑦𝑥 [𝑦 / 𝑥]𝜑𝜑)       𝜑

Theoremsetinds2f 30774* E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑥𝜓    &   (𝑥 = 𝑦 → (𝜑𝜓))    &   (∀𝑦𝑥 𝜓𝜑)       𝜑

Theoremsetinds2 30775* E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (∀𝑦𝑥 𝜓𝜑)       𝜑

20.8.12  Ordinal numbers

Theoremelpotr 30776* A class of transitive sets is partially ordered by E. (Contributed by Scott Fenton, 15-Oct-2010.)
(∀𝑧𝐴 Tr 𝑧 → E Po 𝐴)

Theoremdford5reg 30777 Given ax-reg 8260, an ordinal is a transitive class totally ordered by epsilon. (Contributed by Scott Fenton, 28-Jan-2011.)
(Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))

Theoremdfon2lem1 30778 Lemma for dfon2 30787. (Contributed by Scott Fenton, 28-Feb-2011.)
Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}

Theoremdfon2lem2 30779* Lemma for dfon2 30787. (Contributed by Scott Fenton, 28-Feb-2011.)
{𝑥 ∣ (𝑥𝐴𝜑𝜓)} ⊆ 𝐴

Theoremdfon2lem3 30780* Lemma for dfon2 30787. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)
(𝐴𝑉 → (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (Tr 𝐴 ∧ ∀𝑧𝐴 ¬ 𝑧𝑧)))

Theoremdfon2lem4 30781* Lemma for dfon2 30787. If two sets satisfy the new definition, then one is a subset of the other. (Contributed by Scott Fenton, 25-Feb-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐵𝐴))

Theoremdfon2lem5 30782* Lemma for dfon2 30787. Two sets satisfying the new definition also satisfy trichotomy with respect to . (Contributed by Scott Fenton, 25-Feb-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       ((∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) ∧ ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)) → (𝐴𝐵𝐴 = 𝐵𝐵𝐴))

Theoremdfon2lem6 30783* Lemma for dfon2 30787. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)
((Tr 𝑆 ∧ ∀𝑥𝑆𝑧((𝑧𝑥 ∧ Tr 𝑧) → 𝑧𝑥)) → ∀𝑦((𝑦𝑆 ∧ Tr 𝑦) → 𝑦𝑆))

Theoremdfon2lem7 30784* Lemma for dfon2 30787. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)
𝐴 ∈ V       (∀𝑥((𝑥𝐴 ∧ Tr 𝑥) → 𝑥𝐴) → (𝐵𝐴 → ∀𝑦((𝑦𝐵 ∧ Tr 𝑦) → 𝑦𝐵)))

Theoremdfon2lem8 30785* Lemma for dfon2 30787. The intersection of a nonempty class 𝐴 of new ordinals is itself a new ordinal and is contained within 𝐴 (Contributed by Scott Fenton, 26-Feb-2011.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)) → (∀𝑧((𝑧 𝐴 ∧ Tr 𝑧) → 𝑧 𝐴) ∧ 𝐴𝐴))

Theoremdfon2lem9 30786* Lemma for dfon2 30787. A class of new ordinals is well-founded by E. (Contributed by Scott Fenton, 3-Mar-2011.)
(∀𝑥𝐴𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥) → E Fr 𝐴)

Theoremdfon2 30787* On consists of all sets that contain all its transitive proper subsets. This definition comes from J. R. Isbell, "A Definition of Ordinal Numbers," American Mathematical Monthly, vol 67 (1960), pp. 51-52. (Contributed by Scott Fenton, 20-Feb-2011.)
On = {𝑥 ∣ ∀𝑦((𝑦𝑥 ∧ Tr 𝑦) → 𝑦𝑥)}

Theoremdomep 30788 The domain of the epsilon relation is the universe. (Contributed by Scott Fenton, 27-Oct-2010.)
dom E = V

Theoremrdgprc0 30789 The value of the recursive definition generator at when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)

Theoremrdgprc 30790 The value of the recursive definition generator when 𝐼 is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
𝐼 ∈ V → rec(𝐹, 𝐼) = rec(𝐹, ∅))

Theoremdfrdg2 30791* Alternate definition of the recursive function generator when 𝐼 is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐼𝑉 → rec(𝐹, 𝐼) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))})

Theoremdfrdg3 30792* Generalization of dfrdg2 30791 to remove sethood requirement. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
rec(𝐹, 𝐼) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}

20.8.13  Defined equality axioms

Theoremaxextdfeq 30793 A version of ax-ext 2494 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
𝑧((𝑧𝑥𝑧𝑦) → ((𝑧𝑦𝑧𝑥) → (𝑥𝑤𝑦𝑤)))

Theoremax8dfeq 30794 A version of ax-8 1940 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
𝑧((𝑧𝑥𝑧𝑦) → (𝑤𝑥𝑤𝑦))

Theoremaxextdist 30795 ax-ext 2494 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)
((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧𝑥𝑧𝑦) → 𝑥 = 𝑦))

Theoremaxext4dist 30796 axext4 2498 with distinctors instead of distinct variable restrictions. (Contributed by Scott Fenton, 13-Dec-2010.)
((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 ↔ ∀𝑧(𝑧𝑥𝑧𝑦)))

Theorem19.12b 30797* Version of 19.12vv 2123 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))

Theoremexnel 30798 There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.)
𝑥 ¬ 𝑥𝑦

Theoremdistel 30799 Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 4672 and elirrv 8267.) (Contributed by Scott Fenton, 15-Dec-2010.)
(¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥𝑦)

Theoremaxextndbi 30800 axextnd 9172 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
𝑧(𝑥 = 𝑦 ↔ (𝑧𝑥𝑧𝑦))

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