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Theorem cbvmpt 4023
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 NM, 11-Sep-2011.)
Hypotheses
Ref Expression
cbvmpt.1 𝑦𝐵
cbvmpt.2 𝑥𝐶
cbvmpt.3 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
cbvmpt (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Distinct variable groups:   𝑥,𝐴   𝑦,𝐴
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cbvmpt
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1508 . . . 4 𝑤(𝑥𝐴𝑧 = 𝐵)
2 nfv 1508 . . . . 5 𝑥 𝑤𝐴
3 nfs1v 1912 . . . . 5 𝑥[𝑤 / 𝑥]𝑧 = 𝐵
42, 3nfan 1544 . . . 4 𝑥(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
5 eleq1 2202 . . . . 5 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
6 sbequ12 1744 . . . . 5 (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
75, 6anbi12d 464 . . . 4 (𝑥 = 𝑤 → ((𝑥𝐴𝑧 = 𝐵) ↔ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
81, 4, 7cbvopab1 4001 . . 3 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
9 nfv 1508 . . . . 5 𝑦 𝑤𝐴
10 cbvmpt.1 . . . . . . 7 𝑦𝐵
1110nfeq2 2293 . . . . . 6 𝑦 𝑧 = 𝐵
1211nfsb 1919 . . . . 5 𝑦[𝑤 / 𝑥]𝑧 = 𝐵
139, 12nfan 1544 . . . 4 𝑦(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
14 nfv 1508 . . . 4 𝑤(𝑦𝐴𝑧 = 𝐶)
15 eleq1 2202 . . . . 5 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
16 sbequ 1812 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
17 cbvmpt.2 . . . . . . . 8 𝑥𝐶
1817nfeq2 2293 . . . . . . 7 𝑥 𝑧 = 𝐶
19 cbvmpt.3 . . . . . . . 8 (𝑥 = 𝑦𝐵 = 𝐶)
2019eqeq2d 2151 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = 𝐵𝑧 = 𝐶))
2118, 20sbie 1764 . . . . . 6 ([𝑦 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶)
2216, 21syl6bb 195 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶))
2315, 22anbi12d 464 . . . 4 (𝑤 = 𝑦 → ((𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦𝐴𝑧 = 𝐶)))
2413, 14, 23cbvopab1 4001 . . 3 {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
258, 24eqtri 2160 . 2 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
26 df-mpt 3991 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)}
27 df-mpt 3991 . 2 (𝑦𝐴𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
2825, 26, 273eqtr4i 2170 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  [wsb 1735  wnfc 2268  {copab 3988  cmpt 3989
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-mpt 3991
This theorem is referenced by:  cbvmptv  4024  dffn5imf  5476  fvmpts  5499  fvmpt2  5504  mptfvex  5506  fmptcof  5587  fmptcos  5588  fliftfuns  5699  offval2  5997  qliftfuns  6513  cc2  7087  summodclem2a  11162  zsumdc  11165  fsum3cvg2  11175  cbvprod  11339  cnmpt1t  12468  fsumcncntop  12739  limcmpted  12815
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