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Theorem cbvmptf 5251
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 2377. See cbvmptfg 5252 for a less restrictive version requiring more axioms. (Revised by GG, 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 1914 . . . 4 𝑤(𝑥𝐴𝑧 = 𝐵)
2 cbvmptf.1 . . . . . 6 𝑥𝐴
32nfcri 2897 . . . . 5 𝑥 𝑤𝐴
4 nfs1v 2156 . . . . 5 𝑥[𝑤 / 𝑥]𝑧 = 𝐵
53, 4nfan 1899 . . . 4 𝑥(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
6 eleq1w 2824 . . . . 5 (𝑥 = 𝑤 → (𝑥𝐴𝑤𝐴))
7 sbequ12 2251 . . . . 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 5217 . . 3 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
10 cbvmptf.2 . . . . . 6 𝑦𝐴
1110nfcri 2897 . . . . 5 𝑦 𝑤𝐴
12 cbvmptf.3 . . . . . . 7 𝑦𝐵
1312nfeq2 2923 . . . . . 6 𝑦 𝑧 = 𝐵
1413nfsbv 2330 . . . . 5 𝑦[𝑤 / 𝑥]𝑧 = 𝐵
1511, 14nfan 1899 . . . 4 𝑦(𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
16 nfv 1914 . . . 4 𝑤(𝑦𝐴𝑧 = 𝐶)
17 eleq1w 2824 . . . . 5 (𝑤 = 𝑦 → (𝑤𝐴𝑦𝐴))
18 sbequ 2083 . . . . . 6 (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
19 cbvmptf.4 . . . . . . . 8 𝑥𝐶
2019nfeq2 2923 . . . . . . 7 𝑥 𝑧 = 𝐶
21 cbvmptf.5 . . . . . . . 8 (𝑥 = 𝑦𝐵 = 𝐶)
2221eqeq2d 2748 . . . . . . 7 (𝑥 = 𝑦 → (𝑧 = 𝐵𝑧 = 𝐶))
2320, 22sbiev 2314 . . . . . 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 5217 . . 3 {⟨𝑤, 𝑧⟩ ∣ (𝑤𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
279, 26eqtri 2765 . 2 {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
28 df-mpt 5226 . 2 (𝑥𝐴𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧 = 𝐵)}
29 df-mpt 5226 . 2 (𝑦𝐴𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐶)}
3027, 28, 293eqtr4i 2775 1 (𝑥𝐴𝐵) = (𝑦𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  [wsb 2064  wcel 2108  wnfc 2890  {copab 5205  cmpt 5225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-mpt 5226
This theorem is referenced by:  cbvmpt  5253  resmptf  6057  fvmpt2f  7017  offval2f  7712  suppss2f  32648  fmptdF  32666  acunirnmpt2f  32671  funcnv4mpt  32679  cbvesum  34043  esumpfinvalf  34077  binomcxplemdvbinom  44372  binomcxplemdvsum  44374  binomcxplemnotnn0  44375  fmptff  45276  supxrleubrnmptf  45462  fnlimfv  45678  fnlimfvre2  45692  fnlimf  45693  limsupequzmptf  45746  sge0iunmptlemre  46430  smflim  46792  smflim2  46821  smfsup  46829  smfinf  46833  smflimsuplem2  46836  smflimsuplem5  46839  smflimsup  46843  smfliminf  46846
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