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Theorem finxpeq1 32399
 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.
StepHypRef Expression
1 eleq2 2677 . . . . . . . . . 10 (𝑈 = 𝑉 → (𝑥𝑈𝑥𝑉))
21anbi2d 736 . . . . . . . . 9 (𝑈 = 𝑉 → ((𝑛 = 1𝑜𝑥𝑈) ↔ (𝑛 = 1𝑜𝑥𝑉)))
3 xpeq2 5053 . . . . . . . . . . 11 (𝑈 = 𝑉 → (V × 𝑈) = (V × 𝑉))
43eleq2d 2673 . . . . . . . . . 10 (𝑈 = 𝑉 → (𝑥 ∈ (V × 𝑈) ↔ 𝑥 ∈ (V × 𝑉)))
54ifbid 4058 . . . . . . . . 9 (𝑈 = 𝑉 → if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩) = if(𝑥 ∈ (V × 𝑉), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))
62, 5ifbieq2d 4061 . . . . . . . 8 (𝑈 = 𝑉 → if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)) = if((𝑛 = 1𝑜𝑥𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩)))
76mpt2eq3dv 6619 . . . . . . 7 (𝑈 = 𝑉 → (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))) = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))))
8 rdgeq1 7394 . . . . . . 7 ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))) = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))) → rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩))
97, 8syl 17 . . . . . 6 (𝑈 = 𝑉 → rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩) = rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩))
109fveq1d 6105 . . . . 5 (𝑈 = 𝑉 → (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))
1110eqeq2d 2620 . . . 4 (𝑈 = 𝑉 → (∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)))
1211anbi2d 736 . . 3 (𝑈 = 𝑉 → ((𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))))
1312abbidv 2728 . 2 (𝑈 = 𝑉 → {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))} = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))})
14 df-finxp 32397 . 2 (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑈), ∅, if(𝑥 ∈ (V × 𝑈), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
15 df-finxp 32397 . 2 (𝑉↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1𝑜𝑥𝑉), ∅, if(𝑥 ∈ (V × 𝑉), ⟨ 𝑛, (1st𝑥)⟩, ⟨𝑛, 𝑥⟩))), ⟨𝑁, 𝑦⟩)‘𝑁))}
1613, 14, 153eqtr4g 2669 1 (𝑈 = 𝑉 → (𝑈↑↑𝑁) = (𝑉↑↑𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  Vcvv 3173  ∅c0 3874  ifcif 4036  ⟨cop 4131  ∪ cuni 4372   × cxp 5036  ‘cfv 5804   ↦ cmpt2 6551  ωcom 6957  1st c1st 7057  reccrdg 7392  1𝑜c1o 7440  ↑↑cfinxp 32396 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-finxp 32397 This theorem is referenced by: (None)
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