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Theorem cbvmpo2 41225
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
cbvmpo2.1 𝑦𝐴
cbvmpo2.2 𝑤𝐴
cbvmpo2.3 𝑤𝐶
cbvmpo2.4 𝑦𝐸
cbvmpo2.5 (𝑦 = 𝑤𝐶 = 𝐸)
Assertion
Ref Expression
cbvmpo2 (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 cbvmpo2
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 cbvmpo2.2 . . . . . 6 𝑤𝐴
21nfcri 2975 . . . . 5 𝑤 𝑥𝐴
3 nfcv 2981 . . . . . 6 𝑤𝐵
43nfcri 2975 . . . . 5 𝑤 𝑦𝐵
52, 4nfan 1893 . . . 4 𝑤(𝑥𝐴𝑦𝐵)
6 cbvmpo2.3 . . . . 5 𝑤𝐶
76nfeq2 2999 . . . 4 𝑤 𝑢 = 𝐶
85, 7nfan 1893 . . 3 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
9 cbvmpo2.1 . . . . . 6 𝑦𝐴
109nfcri 2975 . . . . 5 𝑦 𝑥𝐴
11 nfv 1908 . . . . 5 𝑦 𝑤𝐵
1210, 11nfan 1893 . . . 4 𝑦(𝑥𝐴𝑤𝐵)
13 cbvmpo2.4 . . . . 5 𝑦𝐸
1413nfeq2 2999 . . . 4 𝑦 𝑢 = 𝐸
1512, 14nfan 1893 . . 3 𝑦((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)
16 eleq1w 2899 . . . . 5 (𝑦 = 𝑤 → (𝑦𝐵𝑤𝐵))
1716anbi2d 628 . . . 4 (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝐵) ↔ (𝑥𝐴𝑤𝐵)))
18 cbvmpo2.5 . . . . 5 (𝑦 = 𝑤𝐶 = 𝐸)
1918eqeq2d 2835 . . . 4 (𝑦 = 𝑤 → (𝑢 = 𝐶𝑢 = 𝐸))
2017, 19anbi12d 630 . . 3 (𝑦 = 𝑤 → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)))
218, 15, 20cbvoprab2 7235 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)}
22 df-mpo 7156 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
23 df-mpo 7156 . 2 (𝑥𝐴, 𝑤𝐵𝐸) = {⟨⟨𝑥, 𝑤⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑤𝐵) ∧ 𝑢 = 𝐸)}
2421, 22, 233eqtr4i 2858 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑥𝐴, 𝑤𝐵𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2106  wnfc 2965  {coprab 7152  cmpo 7153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-oprab 7155  df-mpo 7156
This theorem is referenced by:  smflimlem4  42913
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