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Theorem cbvmptf 3990
Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Thierry Arnoux, 9-Mar-2017.)
Hypotheses
Ref Expression
cbvmptf.1 𝑥𝐴
cbvmptf.2 𝑦𝐴
cbvmptf.3 𝑦𝐵
cbvmptf.4 𝑥𝐶
cbvmptf.5 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvmptf (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbvmptf
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1491 . . . 4 𝑤(𝑥𝐴𝑧 = 𝐵)
2 cbvmptf.1 . . . . . 6 𝑥𝐴
32nfcri 2250 . . . . 5 𝑥 𝑤𝐴
4 nfs1v 1890 . . . . 5 𝑥[𝑤 / 𝑥]𝑧 = 𝐵
53, 4nfan 1527 . . . 4 𝑥(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
6 eleq1w 2176 . . . . 5 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
7 sbequ12 1727 . . . . 5 (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
86, 7anbi12d 462 . . . 4 (𝑥 = 𝑤 → ((𝑥𝐴𝑧 = 𝐵) ↔ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
91, 5, 8cbvopab1 3969 . . 3 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
10 cbvmptf.2 . . . . . 6 𝑦𝐴
1110nfcri 2250 . . . . 5 𝑦 𝑤𝐴
12 cbvmptf.3 . . . . . . 7 𝑦𝐵
1312nfeq2 2268 . . . . . 6 𝑦 𝑧 = 𝐵
1413nfsb 1897 . . . . 5 𝑦[𝑤 / 𝑥]𝑧 = 𝐵
1511, 14nfan 1527 . . . 4 𝑦(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
16 nfv 1491 . . . 4 𝑤(𝑦𝐴𝑧 = 𝐶)
17 eleq1w 2176 . . . . 5 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
18 cbvmptf.4 . . . . . . 7 𝑥𝐶
1918nfeq2 2268 . . . . . 6 𝑥 𝑧 = 𝐶
20 cbvmptf.5 . . . . . . 7 (𝑥 = 𝑦𝐵 = 𝐶)
2120eqeq2d 2127 . . . . . 6 (𝑥 = 𝑦 → (𝑧 = 𝐵𝑧 = 𝐶))
2219, 21sbhypf 2707 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶))
2317, 22anbi12d 462 . . . 4 (𝑤 = 𝑦 → ((𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦𝐴𝑧 = 𝐶)))
2415, 16, 23cbvopab1 3969 . . 3 {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
259, 24eqtri 2136 . 2 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
26 df-mpt 3959 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)}
27 df-mpt 3959 . 2 (𝑦𝐴𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
2825, 26, 273eqtr4i 2146 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1314  wcel 1463  [wsb 1718  wnfc 2243  {copab 3956  cmpt 3957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-mpt 3959
This theorem is referenced by:  resmptf  4837
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