Theorem cbvmptf 5179

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 5180 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 17-Jan-2024.)

Hypotheses

Ref	Expression
cbvmptf.1	⊢ Ⅎ𝑥𝐴
cbvmptf.2	⊢ Ⅎ𝑦𝐴
cbvmptf.3	⊢ Ⅎ𝑦𝐵
cbvmptf.4	⊢ Ⅎ𝑥𝐶
cbvmptf.5	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

Ref	Expression
cbvmptf	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

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.

Step	Hyp	Ref	Expression
1		nfv 1918	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)
2		cbvmptf.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2893	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
4		nfs1v 2155	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝑧 = 𝐵
5	3, 4	nfan 1903	. . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
6		eleq1w 2821	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
7		sbequ12 2247	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
8	6, 7	anbi12d 630	. . . 4 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
9	1, 5, 8	cbvopab1 5145	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
10		cbvmptf.2	. . . . . 6 ⊢ Ⅎ𝑦𝐴
11	10	nfcri 2893	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
12		cbvmptf.3	. . . . . . 7 ⊢ Ⅎ𝑦𝐵
13	12	nfeq2 2923	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 = 𝐵
14	13	nfsbv 2328	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑧 = 𝐵
15	11, 14	nfan 1903	. . . 4 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
16		nfv 1918	. . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)
17		eleq1w 2821	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
18		sbequ 2087	. . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
19		cbvmptf.4	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
20	19	nfeq2 2923	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝐶
21		cbvmptf.5	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
22	21	eqeq2d 2749	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
23	20, 22	sbiev 2312	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶)
24	18, 23	bitrdi 286	. . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
25	17, 24	anbi12d 630	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)))
26	15, 16, 25	cbvopab1 5145	. . 3 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
27	9, 26	eqtri 2766	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
28		df-mpt 5154	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
29		df-mpt 5154	. 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
30	27, 28, 29	3eqtr4i 2776	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  [wsb 2068   ∈ wcel 2108  Ⅎwnfc 2886  {copab 5132   ↦ cmpt 5153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-mpt 5154

This theorem is referenced by:  cbvmpt  5181  resmptf  5936  fvmpt2f  6858  offval2f  7526  suppss2f  30875  fmptdF  30895  acunirnmpt2f  30900  funcnv4mpt  30908  cbvesum  31910  esumpfinvalf  31944  binomcxplemdvbinom  41860  binomcxplemdvsum  41862  binomcxplemnotnn0  41863  supxrleubrnmptf  42881  fnlimfv  43094  fnlimfvre2  43108  fnlimf  43109  limsupequzmptf  43162  sge0iunmptlemre  43843  smflim  44199  smflim2  44226  smfsup  44234  smfinf  44238  smflimsuplem2  44241  smflimsuplem5  44244  smflimsup  44248  smfliminf  44251