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Theorem cbvmpox 7240
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7241 allows 𝐵 to be a function of 𝑥. (Contributed by NM, 29-Dec-2014.)
Hypotheses
Ref Expression
cbvmpox.1 𝑧𝐵
cbvmpox.2 𝑥𝐷
cbvmpox.3 𝑧𝐶
cbvmpox.4 𝑤𝐶
cbvmpox.5 𝑥𝐸
cbvmpox.6 𝑦𝐸
cbvmpox.7 (𝑥 = 𝑧𝐵 = 𝐷)
cbvmpox.8 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐸)
Assertion
Ref Expression
cbvmpox (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐷𝐸)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐵   𝑦,𝐷
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧,𝑤)   𝐷(𝑥,𝑧,𝑤)   𝐸(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cbvmpox
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 nfv 1908 . . . . 5 𝑧 𝑥𝐴
2 cbvmpox.1 . . . . . 6 𝑧𝐵
32nfcri 2975 . . . . 5 𝑧 𝑦𝐵
41, 3nfan 1893 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
5 cbvmpox.3 . . . . 5 𝑧𝐶
65nfeq2 2999 . . . 4 𝑧 𝑢 = 𝐶
74, 6nfan 1893 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
8 nfv 1908 . . . . 5 𝑤 𝑥𝐴
9 nfcv 2981 . . . . . 6 𝑤𝐵
109nfcri 2975 . . . . 5 𝑤 𝑦𝐵
118, 10nfan 1893 . . . 4 𝑤(𝑥𝐴𝑦𝐵)
12 cbvmpox.4 . . . . 5 𝑤𝐶
1312nfeq2 2999 . . . 4 𝑤 𝑢 = 𝐶
1411, 13nfan 1893 . . 3 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
15 nfv 1908 . . . . 5 𝑥 𝑧𝐴
16 cbvmpox.2 . . . . . 6 𝑥𝐷
1716nfcri 2975 . . . . 5 𝑥 𝑤𝐷
1815, 17nfan 1893 . . . 4 𝑥(𝑧𝐴𝑤𝐷)
19 cbvmpox.5 . . . . 5 𝑥𝐸
2019nfeq2 2999 . . . 4 𝑥 𝑢 = 𝐸
2118, 20nfan 1893 . . 3 𝑥((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)
22 nfv 1908 . . . 4 𝑦(𝑧𝐴𝑤𝐷)
23 cbvmpox.6 . . . . 5 𝑦𝐸
2423nfeq2 2999 . . . 4 𝑦 𝑢 = 𝐸
2522, 24nfan 1893 . . 3 𝑦((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)
26 eleq1w 2899 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2726adantr 481 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐴𝑧𝐴))
28 cbvmpox.7 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝐷)
2928eleq2d 2902 . . . . . 6 (𝑥 = 𝑧 → (𝑦𝐵𝑦𝐷))
30 eleq1w 2899 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐷𝑤𝐷))
3129, 30sylan9bb 510 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝐷))
3227, 31anbi12d 630 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑤𝐷)))
33 cbvmpox.8 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐸)
3433eqeq2d 2836 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑢 = 𝐶𝑢 = 𝐸))
3532, 34anbi12d 630 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)))
367, 14, 21, 25, 35cbvoprab12 7236 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)}
37 df-mpo 7156 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
38 df-mpo 7156 . 2 (𝑧𝐴, 𝑤𝐷𝐸) = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)}
3936, 37, 383eqtr4i 2858 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐷𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  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:  cbvmpo  7241  mpomptsx  7756  dmmpossx  7758  gsumcom2  19018  ptcmpg  22584
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