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Theorem cbvmpo1 41225
Description: Rule to change the first bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
cbvmpo1.1 𝑥𝐵
cbvmpo1.2 𝑧𝐵
cbvmpo1.3 𝑧𝐶
cbvmpo1.4 𝑥𝐸
cbvmpo1.5 (𝑥 = 𝑧𝐶 = 𝐸)
Assertion
Ref Expression
cbvmpo1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑦𝐵𝐸)
Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)   𝐸(𝑥,𝑦,𝑧)

Proof of Theorem cbvmpo1
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 nfv 1908 . . . . 5 𝑧 𝑥𝐴
2 cbvmpo1.2 . . . . . 6 𝑧𝐵
32nfcri 2975 . . . . 5 𝑧 𝑦𝐵
41, 3nfan 1893 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
5 cbvmpo1.3 . . . . 5 𝑧𝐶
65nfeq2 2999 . . . 4 𝑧 𝑢 = 𝐶
74, 6nfan 1893 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
8 nfv 1908 . . . . 5 𝑥 𝑧𝐴
9 cbvmpo1.1 . . . . . 6 𝑥𝐵
109nfcri 2975 . . . . 5 𝑥 𝑦𝐵
118, 10nfan 1893 . . . 4 𝑥(𝑧𝐴𝑦𝐵)
12 cbvmpo1.4 . . . . 5 𝑥𝐸
1312nfeq2 2999 . . . 4 𝑥 𝑢 = 𝐸
1411, 13nfan 1893 . . 3 𝑥((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)
15 eleq1w 2899 . . . . 5 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
1615anbi1d 629 . . . 4 (𝑥 = 𝑧 → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑦𝐵)))
17 cbvmpo1.5 . . . . 5 (𝑥 = 𝑧𝐶 = 𝐸)
1817eqeq2d 2836 . . . 4 (𝑥 = 𝑧 → (𝑢 = 𝐶𝑢 = 𝐸))
1916, 18anbi12d 630 . . 3 (𝑥 = 𝑧 → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)))
207, 14, 19cbvoprab1 7234 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)}
21 df-mpo 7156 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
22 df-mpo 7156 . 2 (𝑧𝐴, 𝑦𝐵𝐸) = {⟨⟨𝑧, 𝑦⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑦𝐵) ∧ 𝑢 = 𝐸)}
2320, 21, 223eqtr4i 2858 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑦𝐵𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  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 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
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 2804  df-cleq 2818  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-opab 5125  df-oprab 7155  df-mpo 7156
This theorem is referenced by: (None)
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