Theorem finxpeq1 37880

Description: Equality theorem for Cartesian exponentiation. (Contributed by ML, 19-Oct-2020.)

Assertion

Ref	Expression
finxpeq1	⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁))

Proof of Theorem finxpeq1

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2851	. . . . . . . . . 10 ⊢ (𝑈 = 𝑉 → (𝑥 ∈ 𝑈 ↔ 𝑥 ∈ 𝑉))
2	1	anbi2d 639	. . . . . . . . 9 ⊢ (𝑈 = 𝑉 → ((𝑛 = 1o ∧ 𝑥 ∈ 𝑈) ↔ (𝑛 = 1o ∧ 𝑥 ∈ 𝑉)))
3		xpeq2 5668	. . . . . . . . . . 11 ⊢ (𝑈 = 𝑉 → (V × 𝑈) = (V × 𝑉))
4	3	eleq2d 2848	. . . . . . . . . 10 ⊢ (𝑈 = 𝑉 → (𝑥 ∈ (V × 𝑈) ↔ 𝑥 ∈ (V × 𝑉)))
5	4	ifbid 4504	. . . . . . . . 9 ⊢ (𝑈 = 𝑉 → if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑥 ∈ (V × 𝑉), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))
6	2, 5	ifbieq2d 4507	. . . . . . . 8 ⊢ (𝑈 = 𝑉 → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if((𝑛 = 1o ∧ 𝑥 ∈ 𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩)))
7	6	mpoeq3dv 7475	. . . . . . 7 ⊢ (𝑈 = 𝑉 → (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))))
8		rdgeq1 8382	. . . . . . 7 ⊢ ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))) → rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩))
9	7, 8	syl 17	. . . . . 6 ⊢ (𝑈 = 𝑉 → rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩))
10	9	fveq1d 6869	. . . . 5 ⊢ (𝑈 = 𝑉 → (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))
11	10	eqeq2d 2773	. . . 4 ⊢ (𝑈 = 𝑉 → (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)))
12	11	anbi2d 639	. . 3 ⊢ (𝑈 = 𝑉 → ((𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))))
13	12	abbidv 2828	. 2 ⊢ (𝑈 = 𝑉 → {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))})
14		df-finxp 37878	. 2 ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
15		df-finxp 37878	. 2 ⊢ (𝑉↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
16	13, 14, 15	3eqtr4g 2822	1 ⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {cab 2740  Vcvv 3454  ∅c0 4285  ifcif 4480  ⟨cop 4588  ∪ cuni 4865   × cxp 5645  ‘cfv 6521   ∈ cmpo 7398  ωcom 7846  1^st c1st 7968  reccrdg 8380  1_oc1o 8430  ↑↑cfinxp 37877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5653  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-iota 6477  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-finxp 37878

This theorem is referenced by: (None)