ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cbvmpt GIF version

Theorem cbvmpt 3910
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 1464 . . . 4 𝑤(𝑥𝐴𝑧 = 𝐵)
2 nfv 1464 . . . . 5 𝑥 𝑤𝐴
3 nfs1v 1860 . . . . 5 𝑥[𝑤 / 𝑥]𝑧 = 𝐵
42, 3nfan 1500 . . . 4 𝑥(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
5 eleq1 2147 . . . . 5 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
6 sbequ12 1698 . . . . 5 (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
75, 6anbi12d 457 . . . 4 (𝑥 = 𝑤 → ((𝑥𝐴𝑧 = 𝐵) ↔ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
81, 4, 7cbvopab1 3888 . . 3 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
9 nfv 1464 . . . . 5 𝑦 𝑤𝐴
10 cbvmpt.1 . . . . . . 7 𝑦𝐵
1110nfeq2 2236 . . . . . 6 𝑦 𝑧 = 𝐵
1211nfsb 1867 . . . . 5 𝑦[𝑤 / 𝑥]𝑧 = 𝐵
139, 12nfan 1500 . . . 4 𝑦(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
14 nfv 1464 . . . 4 𝑤(𝑦𝐴𝑧 = 𝐶)
15 eleq1 2147 . . . . 5 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
16 sbequ 1765 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
17 cbvmpt.2 . . . . . . . 8 𝑥𝐶
1817nfeq2 2236 . . . . . . 7 𝑥 𝑧 = 𝐶
19 cbvmpt.3 . . . . . . . 8 (𝑥 = 𝑦𝐵 = 𝐶)
2019eqeq2d 2096 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = 𝐵𝑧 = 𝐶))
2118, 20sbie 1718 . . . . . 6 ([𝑦 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶)
2216, 21syl6bb 194 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶))
2315, 22anbi12d 457 . . . 4 (𝑤 = 𝑦 → ((𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦𝐴𝑧 = 𝐶)))
2413, 14, 23cbvopab1 3888 . . 3 {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
258, 24eqtri 2105 . 2 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
26 df-mpt 3878 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)}
27 df-mpt 3878 . 2 (𝑦𝐴𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
2825, 26, 273eqtr4i 2115 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436  [wsb 1689  wnfc 2212  {copab 3875  cmpt 3876
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-opab 3877  df-mpt 3878
This theorem is referenced by:  cbvmptv  3911  dffn5imf  5324  fvmpts  5347  fvmpt2  5351  mptfvex  5353  fmptcof  5430  fmptcos  5431  fliftfuns  5540  offval2  5829  qliftfuns  6330  isummolem2a  10684  zisum  10687  fisumcvg2  10696
  Copyright terms: Public domain W3C validator