Theorem finxpeq1 34145

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 2847	. . . . . . . . . 10 ⊢ (𝑈 = 𝑉 → (𝑥 ∈ 𝑈 ↔ 𝑥 ∈ 𝑉))
2	1	anbi2d 620	. . . . . . . . 9 ⊢ (𝑈 = 𝑉 → ((𝑛 = 1o ∧ 𝑥 ∈ 𝑈) ↔ (𝑛 = 1o ∧ 𝑥 ∈ 𝑉)))
3		xpeq2 5424	. . . . . . . . . . 11 ⊢ (𝑈 = 𝑉 → (V × 𝑈) = (V × 𝑉))
4	3	eleq2d 2844	. . . . . . . . . 10 ⊢ (𝑈 = 𝑉 → (𝑥 ∈ (V × 𝑈) ↔ 𝑥 ∈ (V × 𝑉)))
5	4	ifbid 4366	. . . . . . . . 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 4369	. . . . . . . 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 7049	. . . . . . 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 7849	. . . . . . 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 6498	. . . . 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 2781	. . . 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 620	. . 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 2836	. 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 34143	. 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 34143	. 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 2832	1 ⊢ (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1508   ∈ wcel 2051  {cab 2751  Vcvv 3408  ∅c0 4172  ifcif 4344  ⟨cop 4441  ∪ cuni 4708   × cxp 5401  ‘cfv 6185   ∈ cmpo 6976  ωcom 7394  1^st c1st 7497  reccrdg 7847  1_oc1o 7896  ↑↑cfinxp 34142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-xp 5409  df-cnv 5411  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-iota 6149  df-fv 6193  df-oprab 6978  df-mpo 6979  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-finxp 34143

This theorem is referenced by: (None)