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Theorem cbvmpt 3963
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 1476 . . . 4 𝑤(𝑥𝐴𝑧 = 𝐵)
2 nfv 1476 . . . . 5 𝑥 𝑤𝐴
3 nfs1v 1875 . . . . 5 𝑥[𝑤 / 𝑥]𝑧 = 𝐵
42, 3nfan 1512 . . . 4 𝑥(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
5 eleq1 2162 . . . . 5 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
6 sbequ12 1712 . . . . 5 (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
75, 6anbi12d 460 . . . 4 (𝑥 = 𝑤 → ((𝑥𝐴𝑧 = 𝐵) ↔ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
81, 4, 7cbvopab1 3941 . . 3 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
9 nfv 1476 . . . . 5 𝑦 𝑤𝐴
10 cbvmpt.1 . . . . . . 7 𝑦𝐵
1110nfeq2 2252 . . . . . 6 𝑦 𝑧 = 𝐵
1211nfsb 1882 . . . . 5 𝑦[𝑤 / 𝑥]𝑧 = 𝐵
139, 12nfan 1512 . . . 4 𝑦(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
14 nfv 1476 . . . 4 𝑤(𝑦𝐴𝑧 = 𝐶)
15 eleq1 2162 . . . . 5 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
16 sbequ 1779 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
17 cbvmpt.2 . . . . . . . 8 𝑥𝐶
1817nfeq2 2252 . . . . . . 7 𝑥 𝑧 = 𝐶
19 cbvmpt.3 . . . . . . . 8 (𝑥 = 𝑦𝐵 = 𝐶)
2019eqeq2d 2111 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = 𝐵𝑧 = 𝐶))
2118, 20sbie 1732 . . . . . 6 ([𝑦 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶)
2216, 21syl6bb 195 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶))
2315, 22anbi12d 460 . . . 4 (𝑤 = 𝑦 → ((𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦𝐴𝑧 = 𝐶)))
2413, 14, 23cbvopab1 3941 . . 3 {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
258, 24eqtri 2120 . 2 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
26 df-mpt 3931 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)}
27 df-mpt 3931 . 2 (𝑦𝐴𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
2825, 26, 273eqtr4i 2130 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1299  wcel 1448  [wsb 1703  wnfc 2227  {copab 3928  cmpt 3929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-opab 3930  df-mpt 3931
This theorem is referenced by:  cbvmptv  3964  dffn5imf  5408  fvmpts  5431  fvmpt2  5436  mptfvex  5438  fmptcof  5519  fmptcos  5520  fliftfuns  5631  offval2  5928  qliftfuns  6443  summodclem2a  10989  zsumdc  10992  fsum3cvg2  11002  cnmpt1t  12235
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