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Theorem cbvmpt22 38781
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 cbvmpt22.2 . . . . . 6 𝑤𝐴
21nfcri 2755 . . . . 5 𝑤 𝑥𝐴
3 nfcv 2761 . . . . . 6 𝑤𝐵
43nfcri 2755 . . . . 5 𝑤 𝑦𝐵
52, 4nfan 1825 . . . 4 𝑤(𝑥𝐴𝑦𝐵)
6 cbvmpt22.3 . . . . 5 𝑤𝐶
76nfeq2 2776 . . . 4 𝑤 𝑢 = 𝐶
85, 7nfan 1825 . . 3 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
9 cbvmpt22.1 . . . . . 6 𝑦𝐴
109nfcri 2755 . . . . 5 𝑦 𝑥𝐴
11 nfv 1840 . . . . 5 𝑦 𝑤𝐵
1210, 11nfan 1825 . . . 4 𝑦(𝑥𝐴𝑤𝐵)
13 cbvmpt22.4 . . . . 5 𝑦𝐸
1413nfeq2 2776 . . . 4 𝑦 𝑢 = 𝐸
1512, 14nfan 1825 . . 3 𝑦((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)
16 eleq1 2686 . . . . 5 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
1716anbi2d 739 . . . 4 (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝐵) ↔ (𝑥𝐴𝑤𝐵)))
18 cbvmpt22.5 . . . . 5 (𝑦 = 𝑤𝐶 = 𝐸)
1918eqeq2d 2631 . . . 4 (𝑦 = 𝑤 → (𝑢 = 𝐶𝑢 = 𝐸))
2017, 19anbi12d 746 . . 3 (𝑦 = 𝑤 → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)))
218, 15, 20cbvoprab2 6684 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)}
22 df-mpt2 6612 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
23 df-mpt2 6612 . 2 (𝑥𝐴, 𝑤𝐵𝐸) = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)}
2421, 22, 233eqtr4i 2653 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑤𝐵𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wnfc 2748  {coprab 6608  cmpt2 6609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-oprab 6611  df-mpt2 6612
This theorem is referenced by:  smflimlem4  40305
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