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Theorem cbvmptf 5258
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.) (Revised by Thierry Arnoux, 9-Mar-2017.) Add disjoint variable condition to avoid ax-13 2372. See cbvmptfg 5259 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)
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 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1918 . . . 4 𝑤(𝑥𝐴𝑧 = 𝐵)
2 cbvmptf.1 . . . . . 6 𝑥𝐴
32nfcri 2891 . . . . 5 𝑥 𝑤𝐴
4 nfs1v 2154 . . . . 5 𝑥[𝑤 / 𝑥]𝑧 = 𝐵
53, 4nfan 1903 . . . 4 𝑥(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
6 eleq1w 2817 . . . . 5 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
7 sbequ12 2244 . . . . 5 (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
86, 7anbi12d 632 . . . 4 (𝑥 = 𝑤 → ((𝑥𝐴𝑧 = 𝐵) ↔ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
91, 5, 8cbvopab1 5224 . . 3 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
10 cbvmptf.2 . . . . . 6 𝑦𝐴
1110nfcri 2891 . . . . 5 𝑦 𝑤𝐴
12 cbvmptf.3 . . . . . . 7 𝑦𝐵
1312nfeq2 2921 . . . . . 6 𝑦 𝑧 = 𝐵
1413nfsbv 2324 . . . . 5 𝑦[𝑤 / 𝑥]𝑧 = 𝐵
1511, 14nfan 1903 . . . 4 𝑦(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
16 nfv 1918 . . . 4 𝑤(𝑦𝐴𝑧 = 𝐶)
17 eleq1w 2817 . . . . 5 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
18 sbequ 2087 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
19 cbvmptf.4 . . . . . . . 8 𝑥𝐶
2019nfeq2 2921 . . . . . . 7 𝑥 𝑧 = 𝐶
21 cbvmptf.5 . . . . . . . 8 (𝑥 = 𝑦𝐵 = 𝐶)
2221eqeq2d 2744 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = 𝐵𝑧 = 𝐶))
2320, 22sbiev 2309 . . . . . 6 ([𝑦 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶)
2418, 23bitrdi 287 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶))
2517, 24anbi12d 632 . . . 4 (𝑤 = 𝑦 → ((𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦𝐴𝑧 = 𝐶)))
2615, 16, 25cbvopab1 5224 . . 3 {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
279, 26eqtri 2761 . 2 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
28 df-mpt 5233 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)}
29 df-mpt 5233 . 2 (𝑦𝐴𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
3027, 28, 293eqtr4i 2771 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  [wsb 2068  wcel 2107  wnfc 2884  {copab 5211  cmpt 5232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-opab 5212  df-mpt 5233
This theorem is referenced by:  cbvmpt  5260  resmptf  6040  fvmpt2f  7000  offval2f  7685  suppss2f  31863  fmptdF  31881  acunirnmpt2f  31886  funcnv4mpt  31894  cbvesum  33040  esumpfinvalf  33074  binomcxplemdvbinom  43112  binomcxplemdvsum  43114  binomcxplemnotnn0  43115  fmptff  43974  supxrleubrnmptf  44161  fnlimfv  44379  fnlimfvre2  44393  fnlimf  44394  limsupequzmptf  44447  sge0iunmptlemre  45131  smflim  45493  smflim2  45522  smfsup  45530  smfinf  45534  smflimsuplem2  45537  smflimsuplem5  45540  smflimsup  45544  smfliminf  45547
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