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Theorem cbvmpox 5943
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 5944 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 1526 . . . . 5 𝑧 𝑥𝐴
2 cbvmpox.1 . . . . . 6 𝑧𝐵
32nfcri 2311 . . . . 5 𝑧 𝑦𝐵
41, 3nfan 1563 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
5 cbvmpox.3 . . . . 5 𝑧𝐶
65nfeq2 2329 . . . 4 𝑧 𝑢 = 𝐶
74, 6nfan 1563 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
8 nfv 1526 . . . . 5 𝑤 𝑥𝐴
9 nfcv 2317 . . . . . 6 𝑤𝐵
109nfcri 2311 . . . . 5 𝑤 𝑦𝐵
118, 10nfan 1563 . . . 4 𝑤(𝑥𝐴𝑦𝐵)
12 cbvmpox.4 . . . . 5 𝑤𝐶
1312nfeq2 2329 . . . 4 𝑤 𝑢 = 𝐶
1411, 13nfan 1563 . . 3 𝑤((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)
15 nfv 1526 . . . . 5 𝑥 𝑧𝐴
16 cbvmpox.2 . . . . . 6 𝑥𝐷
1716nfcri 2311 . . . . 5 𝑥 𝑤𝐷
1815, 17nfan 1563 . . . 4 𝑥(𝑧𝐴𝑤𝐷)
19 cbvmpox.5 . . . . 5 𝑥𝐸
2019nfeq2 2329 . . . 4 𝑥 𝑢 = 𝐸
2118, 20nfan 1563 . . 3 𝑥((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)
22 nfv 1526 . . . 4 𝑦(𝑧𝐴𝑤𝐷)
23 cbvmpox.6 . . . . 5 𝑦𝐸
2423nfeq2 2329 . . . 4 𝑦 𝑢 = 𝐸
2522, 24nfan 1563 . . 3 𝑦((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)
26 eleq1 2238 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2726adantr 276 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑥𝐴𝑧𝐴))
28 cbvmpox.7 . . . . . . 7 (𝑥 = 𝑧𝐵 = 𝐷)
2928eleq2d 2245 . . . . . 6 (𝑥 = 𝑧 → (𝑦𝐵𝑦𝐷))
30 eleq1 2238 . . . . . 6 (𝑦 = 𝑤 → (𝑦𝐷𝑤𝐷))
3129, 30sylan9bb 462 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑦𝐵𝑤𝐷))
3227, 31anbi12d 473 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → ((𝑥𝐴𝑦𝐵) ↔ (𝑧𝐴𝑤𝐷)))
33 cbvmpox.8 . . . . 5 ((𝑥 = 𝑧𝑦 = 𝑤) → 𝐶 = 𝐸)
3433eqeq2d 2187 . . . 4 ((𝑥 = 𝑧𝑦 = 𝑤) → (𝑢 = 𝐶𝑢 = 𝐸))
3532, 34anbi12d 473 . . 3 ((𝑥 = 𝑧𝑦 = 𝑤) → (((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶) ↔ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)))
367, 14, 21, 25, 35cbvoprab12 5939 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)} = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)}
37 df-mpo 5870 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑢⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑢 = 𝐶)}
38 df-mpo 5870 . 2 (𝑧𝐴, 𝑤𝐷𝐸) = {⟨⟨𝑧, 𝑤⟩, 𝑢⟩ ∣ ((𝑧𝐴𝑤𝐷) ∧ 𝑢 = 𝐸)}
3936, 37, 383eqtr4i 2206 1 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑧𝐴, 𝑤𝐷𝐸)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2146  wnfc 2304  {coprab 5866  cmpo 5867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-opab 4060  df-oprab 5869  df-mpo 5870
This theorem is referenced by:  cbvmpo  5944  mpomptsx  6188  dmmpossx  6190
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