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Theorem cbvmpt22 38305
 Description: Rule to change the second bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
cbvmpt22.1 𝑦𝐴
cbvmpt22.2 𝑤𝐴
cbvmpt22.3 𝑤𝐶
cbvmpt22.4 𝑦𝐸
cbvmpt22.5 (𝑦 = 𝑤𝐶 = 𝐸)
Assertion
Ref Expression
cbvmpt22 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑤𝐵𝐸)
Distinct variable groups:   𝑤,𝐵,𝑦   𝑥,𝑤,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑤)   𝐵(𝑥)   𝐶(𝑥,𝑦,𝑤)   𝐸(𝑥,𝑦,𝑤)

Proof of Theorem cbvmpt22
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 nfcv 2751 . . . . . 6 𝑤𝑥
2 cbvmpt22.2 . . . . . 6 𝑤𝐴
31, 2nfel 2763 . . . . 5 𝑤 𝑥𝐴
4 nfcv 2751 . . . . . 6 𝑤𝐵
54nfcri 2745 . . . . 5 𝑤 𝑦𝐵
63, 5nfan 1816 . . . 4 𝑤(𝑥𝐴𝑦𝐵)
7 cbvmpt22.3 . . . . 5 𝑤𝐶
87nfeq2 2766 . . . 4 𝑤 𝑢 = 𝐶
96, 8nfan 1816 . . 3 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
10 nfcv 2751 . . . . . 6 𝑦𝑥
11 cbvmpt22.1 . . . . . 6 𝑦𝐴
1210, 11nfel 2763 . . . . 5 𝑦 𝑥𝐴
13 nfv 1830 . . . . 5 𝑦 𝑤𝐵
1412, 13nfan 1816 . . . 4 𝑦(𝑥𝐴𝑤𝐵)
15 cbvmpt22.4 . . . . 5 𝑦𝐸
1615nfeq2 2766 . . . 4 𝑦 𝑢 = 𝐸
1714, 16nfan 1816 . . 3 𝑦((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)
18 eleq1 2676 . . . . 5 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
1918anbi2d 736 . . . 4 (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝐵) ↔ (𝑥𝐴𝑤𝐵)))
20 cbvmpt22.5 . . . . 5 (𝑦 = 𝑤𝐶 = 𝐸)
2120eqeq2d 2620 . . . 4 (𝑦 = 𝑤 → (𝑢 = 𝐶𝑢 = 𝐸))
2219, 21anbi12d 743 . . 3 (𝑦 = 𝑤 → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)))
239, 17, 22cbvoprab2 6626 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)}
24 df-mpt2 6554 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
25 df-mpt2 6554 . 2 (𝑥𝐴, 𝑤𝐵𝐸) = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)}
2623, 24, 253eqtr4i 2642 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑤𝐵𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  {coprab 6550   ↦ cmpt2 6551 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-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-oprab 6553  df-mpt2 6554 This theorem is referenced by:  smflimlem4  39660
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