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Theorem cbvmptf 5130
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 2371. See cbvmptfg 5131 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 1920 . . . 4 𝑤(𝑥𝐴𝑧 = 𝐵)
2 cbvmptf.1 . . . . . 6 𝑥𝐴
32nfcri 2886 . . . . 5 𝑥 𝑤𝐴
4 nfs1v 2160 . . . . 5 𝑥[𝑤 / 𝑥]𝑧 = 𝐵
53, 4nfan 1905 . . . 4 𝑥(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
6 eleq1w 2815 . . . . 5 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
7 sbequ12 2252 . . . . 5 (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
86, 7anbi12d 634 . . . 4 (𝑥 = 𝑤 → ((𝑥𝐴𝑧 = 𝐵) ↔ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
91, 5, 8cbvopab1 5104 . . 3 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
10 cbvmptf.2 . . . . . 6 𝑦𝐴
1110nfcri 2886 . . . . 5 𝑦 𝑤𝐴
12 cbvmptf.3 . . . . . . 7 𝑦𝐵
1312nfeq2 2916 . . . . . 6 𝑦 𝑧 = 𝐵
1413nfsbv 2331 . . . . 5 𝑦[𝑤 / 𝑥]𝑧 = 𝐵
1511, 14nfan 1905 . . . 4 𝑦(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
16 nfv 1920 . . . 4 𝑤(𝑦𝐴𝑧 = 𝐶)
17 eleq1w 2815 . . . . 5 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
18 sbequ 2092 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
19 cbvmptf.4 . . . . . . . 8 𝑥𝐶
2019nfeq2 2916 . . . . . . 7 𝑥 𝑧 = 𝐶
21 cbvmptf.5 . . . . . . . 8 (𝑥 = 𝑦𝐵 = 𝐶)
2221eqeq2d 2749 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = 𝐵𝑧 = 𝐶))
2320, 22sbiev 2315 . . . . . 6 ([𝑦 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶)
2418, 23bitrdi 290 . . . . 5 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵𝑧 = 𝐶))
2517, 24anbi12d 634 . . . 4 (𝑤 = 𝑦 → ((𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦𝐴𝑧 = 𝐶)))
2615, 16, 25cbvopab1 5104 . . 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 5112 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)}
29 df-mpt 5112 . 2 (𝑦𝐴𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
3027, 28, 293eqtr4i 2771 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  [wsb 2073  wcel 2113  wnfc 2879  {copab 5093  cmpt 5111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3400  df-un 3849  df-sn 4518  df-pr 4520  df-op 4524  df-opab 5094  df-mpt 5112
This theorem is referenced by:  cbvmpt  5132  resmptf  5882  fvmpt2f  6777  offval2f  7440  suppss2f  30548  fmptdF  30568  acunirnmpt2f  30573  funcnv4mpt  30581  cbvesum  31580  esumpfinvalf  31614  binomcxplemdvbinom  41501  binomcxplemdvsum  41503  binomcxplemnotnn0  41504  supxrleubrnmptf  42523  fnlimfv  42738  fnlimfvre2  42752  fnlimf  42753  limsupequzmptf  42806  sge0iunmptlemre  43487  smflim  43843  smflim2  43870  smfsup  43878  smfinf  43882  smflimsuplem2  43885  smflimsuplem5  43888  smflimsup  43892  smfliminf  43895
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