Theorem finxpeq2 37375

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

Assertion

Ref	Expression
finxpeq2	⊢ (𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁))

Proof of Theorem finxpeq2

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

Step	Hyp	Ref	Expression
1		eleq1 2816	. . . 4 ⊢ (𝑀 = 𝑁 → (𝑀 ∈ ω ↔ 𝑁 ∈ ω))
2		opeq1 4837	. . . . . . 7 ⊢ (𝑀 = 𝑁 → ⟨𝑀, 𝑦⟩ = ⟨𝑁, 𝑦⟩)
3		rdgeq2 8380	. . . . . . 7 ⊢ (⟨𝑀, 𝑦⟩ = ⟨𝑁, 𝑦⟩ → rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑀, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩))
4	2, 3	syl 17	. . . . . 6 ⊢ (𝑀 = 𝑁 → rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑀, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩))
5		id 22	. . . . . 6 ⊢ (𝑀 = 𝑁 → 𝑀 = 𝑁)
6	4, 5	fveq12d 6865	. . . . 5 ⊢ (𝑀 = 𝑁 → (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑀, 𝑦⟩)‘𝑀) = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))
7	6	eqeq2d 2740	. . . 4 ⊢ (𝑀 = 𝑁 → (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑀, 𝑦⟩)‘𝑀) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)))
8	1, 7	anbi12d 632	. . 3 ⊢ (𝑀 = 𝑁 → ((𝑀 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑀, 𝑦⟩)‘𝑀)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))))
9	8	abbidv 2795	. 2 ⊢ (𝑀 = 𝑁 → {𝑦 ∣ (𝑀 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑀, 𝑦⟩)‘𝑀))} = {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))})
10		df-finxp 37372	. 2 ⊢ (𝑈↑↑𝑀) = {𝑦 ∣ (𝑀 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨∪ 𝑛, (1st ‘𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑀, 𝑦⟩)‘𝑀))}
11		df-finxp 37372	. 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))}
12	9, 10, 11	3eqtr4g 2789	1 ⊢ (𝑀 = 𝑁 → (𝑈↑↑𝑀) = (𝑈↑↑𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {cab 2707  Vcvv 3447  ∅c0 4296  ifcif 4488  ⟨cop 4595  ∪ cuni 4871   × cxp 5636  ‘cfv 6511   ∈ cmpo 7389  ωcom 7842  1^st c1st 7966  reccrdg 8377  1_oc1o 8427  ↑↑cfinxp 37371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-xp 5644  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-iota 6464  df-fv 6519  df-ov 7390  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-finxp 37372

This theorem is referenced by:  finxp2o  37387  finxp3o  37388  finxp00  37390