P. 344 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 34301-34400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	supfz 34301	The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁)
Theorem	inffz 34302	The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by AV, 10-Oct-2021.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀)
Theorem	fz0n 34303	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))
Theorem	shftvalg 34304	Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴)))
Theorem	divcnvlin 34305*	Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((𝑘 + 𝐴) / (𝑘 + 𝐵)))    ⇒   ⊢ (𝜑 → 𝐹 ⇝ 1)
Theorem	climlec3 34306*	Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐵)
Theorem	logi 34307	Calculate the logarithm of i. (Contributed by Scott Fenton, 13-Apr-2020.)
⊢ (log‘i) = (i · (π / 2))
Theorem	iexpire 34308	i raised to itself is real. (Contributed by Scott Fenton, 13-Apr-2020.)
⊢ (i↑𝑐i) ∈ ℝ
Theorem	bcneg1 34309	The binomial coefficent over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0)
Theorem	bcm1nt 34310	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 / (𝑁 − 𝐾))))
Theorem	bcprod 34311*	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)))
Theorem	bccolsum 34312*	A column-sum rule for binomial coefficents. (Contributed by Scott Fenton, 24-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1)))
21.9.6  Infinite products
Theorem	iprodefisumlem 34313	Lemma for iprodefisum 34314. (Contributed by Scott Fenton, 11-Feb-2018.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)    ⇒   ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))
Theorem	iprodefisum 34314*	Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵))
Theorem	iprodgam 34315*	An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.)
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    ⇒   ⊢ (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴))
21.9.7  Factorial limits
Theorem	faclimlem1 34316*	Lemma for faclim 34319. 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))))))
Theorem	faclimlem2 34317*	Lemma for faclim 34319. 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))
Theorem	faclimlem3 34318	Lemma for faclim 34319. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝐵 ∈ 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)))))
Theorem	faclim 34319*	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))
Theorem	iprodfac 34320*	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))))
Theorem	faclim2 34321*	Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((!‘𝑛) · ((𝑛 + 1)↑𝑀)) / (!‘(𝑛 + 𝑀))))    ⇒   ⊢ (𝑀 ∈ ℕ0 → 𝐹 ⇝ 1)
21.9.8  Greatest common divisor and divisibility
Theorem	gcd32 34322	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))
Theorem	gcdabsorb 34323	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))
21.9.9  Properties of relationships
Theorem	dftr6 34324	A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
Theorem	coep 34325*	Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥)
Theorem	coepr 34326*	Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵)
Theorem	dffr5 34327	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)))
Theorem	dfso2 34328	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))))
Theorem	br8 34329*	Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁))    &   ⊢ (𝑔 = 𝐺 → (𝜁 ↔ 𝜎))    &   ⊢ (ℎ = 𝐻 → (𝜎 ↔ 𝜌))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)}    ⇒   ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜌))
Theorem	br6 34330*	Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 (𝑝 = ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ ∧ 𝑞 = ⟨𝑑, ⟨𝑒, 𝑓⟩⟩ ∧ 𝜑)}    ⇒   ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄) ∧ (𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄)) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩𝑅⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ 𝜁))
Theorem	br4 34331*	Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)}    ⇒   ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ 𝜏))
Theorem	cnvco1 34332	Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴)
Theorem	cnvco2 34333	Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴)
Theorem	eldm3 34334	Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.)
⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅)
Theorem	elrn3 34335	Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.)
⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅)
Theorem	pocnv 34336	The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴)
Theorem	socnv 34337	The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴)
Theorem	sotrd 34338	Transitivity law for strict orderings, deduction form. (Contributed by Scott Fenton, 24-Nov-2021.)
⊢ (𝜑 → 𝑅 Or 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑋𝑅𝑌)    &   ⊢ (𝜑 → 𝑌𝑅𝑍)    ⇒   ⊢ (𝜑 → 𝑋𝑅𝑍)
Theorem	elintfv 34339*	Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.)
⊢ 𝑋 ∈ V    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))
21.9.10  Properties of functions and mappings
Theorem	funpsstri 34340	A condition for subset trichotomy for functions. (Contributed by Scott Fenton, 19-Apr-2011.)
⊢ ((Fun 𝐻 ∧ (𝐹 ⊆ 𝐻 ∧ 𝐺 ⊆ 𝐻) ∧ (dom 𝐹 ⊆ dom 𝐺 ∨ dom 𝐺 ⊆ dom 𝐹)) → (𝐹 ⊊ 𝐺 ∨ 𝐹 = 𝐺 ∨ 𝐺 ⊊ 𝐹))
Theorem	fundmpss 34341	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 → (𝐹 ⊊ 𝐺 → dom 𝐹 ⊊ dom 𝐺))
Theorem	funsseq 34342	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 = 𝐺 ↔ 𝐹 ⊆ 𝐺))
Theorem	fununiq 34343	The uniqueness condition of functions. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (Fun 𝐹 → ((𝐴𝐹𝐵 ∧ 𝐴𝐹𝐶) → 𝐵 = 𝐶))
Theorem	funbreq 34344	An equality condition for functions. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → (𝐴𝐹𝐶 ↔ 𝐵 = 𝐶))
Theorem	br1steq 34345	Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩1st 𝐶 ↔ 𝐶 = 𝐴)
Theorem	br2ndeq 34346	Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩2nd 𝐶 ↔ 𝐶 = 𝐵)
Theorem	dfdm5 34347	Definition of domain in terms of 1st and image. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ dom 𝐴 = ((1st ↾ (V × V)) “ 𝐴)
Theorem	dfrn5 34348	Definition of range in terms of 2nd and image. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ ran 𝐴 = ((2nd ↾ (V × V)) “ 𝐴)
Theorem	opelco3 34349	Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.)
⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
Theorem	elima4 34350	Quantifier-free expression saying that a class is a member of an image. (Contributed by Scott Fenton, 8-May-2018.)
⊢ (𝐴 ∈ (𝑅 “ 𝐵) ↔ (𝑅 ∩ (𝐵 × {𝐴})) ≠ ∅)
Theorem	fv1stcnv 34351	The value of the converse of 1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑉) → (◡(1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)
Theorem	fv2ndcnv 34352	The value of the converse of 2nd restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → (◡(2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)
21.9.11  Set induction (or epsilon induction)
Theorem	setinds 34353*	Principle of set induction (or E-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.)
⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥]𝜑 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	setinds2f 34354*	E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)    ⇒   ⊢ 𝜑
Theorem	setinds2 34355*	E induction schema, using implicit substitution. (Contributed by Scott Fenton, 10-Mar-2011.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    &   ⊢ (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)    ⇒   ⊢ 𝜑
21.9.12  Ordinal numbers
Theorem	elpotr 34356*	A class of transitive sets is partially ordered by E. (Contributed by Scott Fenton, 15-Oct-2010.)
⊢ (∀𝑧 ∈ 𝐴 Tr 𝑧 → E Po 𝐴)
Theorem	dford5reg 34357	Given ax-reg 9528, an ordinal is a transitive class totally ordered by the membership relation. (Contributed by Scott Fenton, 28-Jan-2011.)
⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E Or 𝐴))
Theorem	dfon2lem1 34358	Lemma for dfon2 34367. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}
Theorem	dfon2lem2 34359*	Lemma for dfon2 34367. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ ∪ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓)} ⊆ 𝐴
Theorem	dfon2lem3 34360*	Lemma for dfon2 34367. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)
⊢ (𝐴 ∈ 𝑉 → (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧)))
Theorem	dfon2lem4 34361*	Lemma for dfon2 34367. 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((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
Theorem	dfon2lem5 34362*	Lemma for dfon2 34367. Two sets satisfying the new definition also satisfy trichotomy with respect to ∈. (Contributed by Scott Fenton, 25-Feb-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ∧ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
Theorem	dfon2lem6 34363*	Lemma for dfon2 34367. A transitive class of sets satisfying the new definition satisfies the new definition. (Contributed by Scott Fenton, 25-Feb-2011.)
⊢ ((Tr 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑧((𝑧 ⊊ 𝑥 ∧ Tr 𝑧) → 𝑧 ∈ 𝑥)) → ∀𝑦((𝑦 ⊊ 𝑆 ∧ Tr 𝑦) → 𝑦 ∈ 𝑆))
Theorem	dfon2lem7 34364*	Lemma for dfon2 34367. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (𝐵 ∈ 𝐴 → ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)))
Theorem	dfon2lem8 34365*	Lemma for dfon2 34367. 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 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)) → (∀𝑧((𝑧 ⊊ ∩ 𝐴 ∧ Tr 𝑧) → 𝑧 ∈ ∩ 𝐴) ∧ ∩ 𝐴 ∈ 𝐴))
Theorem	dfon2lem9 34366*	Lemma for dfon2 34367. A class of new ordinals is well-founded by E. (Contributed by Scott Fenton, 3-Mar-2011.)
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥) → E Fr 𝐴)
Theorem	dfon2 34367*	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((𝑦 ⊊ 𝑥 ∧ Tr 𝑦) → 𝑦 ∈ 𝑥)}
Theorem	rdgprc0 34368	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)‘∅) = ∅)
Theorem	rdgprc 34369	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, ∅))
Theorem	dfrdg2 34370*	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.)
⊢ (𝐼 ∈ 𝑉 → rec(𝐹, 𝐼) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))})
Theorem	dfrdg3 34371*	Generalization of dfrdg2 34370 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 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐼 ∈ V, 𝐼, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
21.9.13  Defined equality axioms
Theorem	axextdfeq 34372	A version of ax-ext 2707 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → ((𝑧 ∈ 𝑦 → 𝑧 ∈ 𝑥) → (𝑥 ∈ 𝑤 → 𝑦 ∈ 𝑤)))
Theorem	ax8dfeq 34373	A version of ax-8 2108 for use with defined equality. (Contributed by Scott Fenton, 12-Dec-2010.)
⊢ ∃𝑧((𝑧 ∈ 𝑥 → 𝑧 ∈ 𝑦) → (𝑤 ∈ 𝑥 → 𝑤 ∈ 𝑦))
Theorem	axextdist 34374	ax-ext 2707 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))
Theorem	axextbdist 34375	axextb 2710 with distinctors instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ((¬ ∀𝑧 𝑧 = 𝑥 ∧ ¬ ∀𝑧 𝑧 = 𝑦) → (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦)))
Theorem	19.12b 34376*	Version of 19.12vv 2343 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))
Theorem	exnel 34377	There is always a set not in 𝑦. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ∃𝑥 ¬ 𝑥 ∈ 𝑦
Theorem	distel 34378	Distinctors in terms of membership. (NOTE: this only works with relations where we can prove el 5394 and elirrv 9532.) (Contributed by Scott Fenton, 15-Dec-2010.)
⊢ (¬ ∀𝑦 𝑦 = 𝑥 ↔ ¬ ∀𝑦 ¬ 𝑥 ∈ 𝑦)
Theorem	axextndbi 34379	axextnd 10527 as a biconditional. (Contributed by Scott Fenton, 14-Dec-2010.)
⊢ ∃𝑧(𝑥 = 𝑦 ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))
21.9.14  Hypothesis builders
Theorem	hbntg 34380	A more general form of hbnt 2290. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (∀𝑥(𝜑 → ∀𝑥𝜓) → (¬ 𝜓 → ∀𝑥 ¬ 𝜑))
Theorem	hbimtg 34381	A more general and closed form of hbim 2295. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ ((∀𝑥(𝜑 → ∀𝑥𝜒) ∧ (𝜓 → ∀𝑥𝜃)) → ((𝜒 → 𝜓) → ∀𝑥(𝜑 → 𝜃)))
Theorem	hbaltg 34382	A more general and closed form of hbal 2167. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (∀𝑥(𝜑 → ∀𝑦𝜓) → (∀𝑥𝜑 → ∀𝑦∀𝑥𝜓))
Theorem	hbng 34383	A more general form of hbn 2291. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (𝜑 → ∀𝑥𝜓)    ⇒   ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜑)
Theorem	hbimg 34384	A more general form of hbim 2295. (Contributed by Scott Fenton, 13-Dec-2010.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜒 → ∀𝑥𝜃)    ⇒   ⊢ ((𝜓 → 𝜒) → ∀𝑥(𝜑 → 𝜃))
21.9.15  Well-founded zero, successor, and limits
Syntax	cwsuc 34385	Declare the syntax for well-founded successor.
class wsuc(𝑅, 𝐴, 𝑋)
Syntax	cwlim 34386	Declare the syntax for well-founded limit class.
class WLim(𝑅, 𝐴)
Definition	df-wsuc 34387	Define the concept of a successor in a well-founded set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ wsuc(𝑅, 𝐴, 𝑋) = inf(Pred(◡𝑅, 𝐴, 𝑋), 𝐴, 𝑅)
Definition	df-wlim 34388*	Define the class of limit points of a well-founded set. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ WLim(𝑅, 𝐴) = {𝑥 ∈ 𝐴 ∣ (𝑥 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑥 = sup(Pred(𝑅, 𝐴, 𝑥), 𝐴, 𝑅))}
Theorem	wsuceq123 34389	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝑋 = 𝑌) → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐵, 𝑌))
Theorem	wsuceq1 34390	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑅 = 𝑆 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑆, 𝐴, 𝑋))
Theorem	wsuceq2 34391	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝐴 = 𝐵 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐵, 𝑋))
Theorem	wsuceq3 34392	Equality theorem for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.)
⊢ (𝑋 = 𝑌 → wsuc(𝑅, 𝐴, 𝑋) = wsuc(𝑅, 𝐴, 𝑌))
Theorem	nfwsuc 34393	Bound-variable hypothesis builder for well-founded successor. (Contributed by Scott Fenton, 13-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝑋    ⇒   ⊢ Ⅎ𝑥wsuc(𝑅, 𝐴, 𝑋)
Theorem	wlimeq12 34394	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐵))
Theorem	wlimeq1 34395	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
⊢ (𝑅 = 𝑆 → WLim(𝑅, 𝐴) = WLim(𝑆, 𝐴))
Theorem	wlimeq2 34396	Equality theorem for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.)
⊢ (𝐴 = 𝐵 → WLim(𝑅, 𝐴) = WLim(𝑅, 𝐵))
Theorem	nfwlim 34397	Bound-variable hypothesis builder for the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Proof shortened by AV, 10-Oct-2021.)
⊢ Ⅎ𝑥𝑅    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥WLim(𝑅, 𝐴)
Theorem	elwlim 34398	Membership in the limit class. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ (𝑋 ∈ WLim(𝑅, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ≠ inf(𝐴, 𝐴, 𝑅) ∧ 𝑋 = sup(Pred(𝑅, 𝐴, 𝑋), 𝐴, 𝑅)))
Theorem	wzel 34399	The zero of a well-founded set is a member of that set. (Contributed by Scott Fenton, 13-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐴 ≠ ∅) → inf(𝐴, 𝐴, 𝑅) ∈ 𝐴)
Theorem	wsuclem 34400*	Lemma for the supremum properties of well-founded successor. (Contributed by Scott Fenton, 15-Jun-2018.) (Revised by AV, 10-Oct-2021.)
⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑅 Se 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → ∃𝑤 ∈ 𝐴 𝑋𝑅𝑤)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ Pred (◡𝑅, 𝐴, 𝑋) ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ Pred (◡𝑅, 𝐴, 𝑋)𝑧𝑅𝑦)))