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Theorem cbvmpox 7255
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7256 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 1920 . . . . 5 𝑧 𝑥𝐴
2 cbvmpox.1 . . . . . 6 𝑧𝐵
32nfcri 2886 . . . . 5 𝑧 𝑦𝐵
41, 3nfan 1905 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
5 cbvmpox.3 . . . . 5 𝑧𝐶
65nfeq2 2916 . . . 4 𝑧 𝑢 = 𝐶
74, 6nfan 1905 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
8 nfv 1920 . . . . 5 𝑤 𝑥𝐴
9 nfcv 2899 . . . . . 6 𝑤𝐵
109nfcri 2886 . . . . 5 𝑤 𝑦𝐵
118, 10nfan 1905 . . . 4 𝑤(𝑥𝐴𝑦𝐵)
12 cbvmpox.4 . . . . 5 𝑤𝐶
1312nfeq2 2916 . . . 4 𝑤 𝑢 = 𝐶
1411, 13nfan 1905 . . 3 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
15 nfv 1920 . . . . 5 𝑥 𝑧𝐴
16 cbvmpox.2 . . . . . 6 𝑥𝐷
1716nfcri 2886 . . . . 5 𝑥 𝑤𝐷
1815, 17nfan 1905 . . . 4 𝑥(𝑧𝐴𝑤𝐷)
19 cbvmpox.5 . . . . 5 𝑥𝐸
2019nfeq2 2916 . . . 4 𝑥 𝑢 = 𝐸
2118, 20nfan 1905 . . 3 𝑥((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)
22 nfv 1920 . . . 4 𝑦(𝑧𝐴𝑤𝐷)
23 cbvmpox.6 . . . . 5 𝑦𝐸
2423nfeq2 2916 . . . 4 𝑦 𝑢 = 𝐸
2522, 24nfan 1905 . . 3 𝑦((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)
26 eleq1w 2815 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2726adantr 484 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐴𝑧𝐴))
28 cbvmpox.7 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝐷)
2928eleq2d 2818 . . . . . 6 (𝑥 = 𝑧 → (𝑦𝐵𝑦𝐷))
30 eleq1w 2815 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐷𝑤𝐷))
3129, 30sylan9bb 513 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝐷))
3227, 31anbi12d 634 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑤𝐷)))
33 cbvmpox.8 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐸)
3433eqeq2d 2749 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑢 = 𝐶𝑢 = 𝐸))
3532, 34anbi12d 634 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)))
367, 14, 21, 25, 35cbvoprab12 7251 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)}
37 df-mpo 7169 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
38 df-mpo 7169 . 2 (𝑧𝐴, 𝑤𝐷𝐸) = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)}
3936, 37, 383eqtr4i 2771 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐷𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  wnfc 2879  {coprab 7165  cmpo 7166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3399  df-dif 3844  df-un 3846  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-opab 5090  df-oprab 7168  df-mpo 7169
This theorem is referenced by:  cbvmpo  7256  mpomptsx  7780  dmmpossx  7782  gsumcom2  19207  ptcmpg  22801
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