Theorem cbvmptf 5184

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 2373. See cbvmptfg 5185 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 2895	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
4		nfs1v 2154	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝑧 = 𝐵
5	3, 4	nfan 1903	. . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
6		eleq1w 2822	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
7		sbequ12 2245	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
8	6, 7	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
9	1, 5, 8	cbvopab1 5150	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
10		cbvmptf.2	. . . . . 6 ⊢ Ⅎ𝑦𝐴
11	10	nfcri 2895	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
12		cbvmptf.3	. . . . . . 7 ⊢ Ⅎ𝑦𝐵
13	12	nfeq2 2925	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 = 𝐵
14	13	nfsbv 2325	. . . . 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 2822	. . . . 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 2925	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝐶
21		cbvmptf.5	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
22	21	eqeq2d 2750	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
23	20, 22	sbiev 2310	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶)
24	18, 23	bitrdi 287	. . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
25	17, 24	anbi12d 631	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)))
26	15, 16, 25	cbvopab1 5150	. . 3 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
27	9, 26	eqtri 2767	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
28		df-mpt 5159	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
29		df-mpt 5159	. 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
30	27, 28, 29	3eqtr4i 2777	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539  [wsb 2068   ∈ wcel 2107  Ⅎwnfc 2888  {copab 5137   ↦ cmpt 5158

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 2710

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-rab 3074  df-v 3435  df-dif 3891  df-un 3893  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-opab 5138  df-mpt 5159

This theorem is referenced by:  cbvmpt  5186  resmptf  5950  fvmpt2f  6885  offval2f  7557  suppss2f  30983  fmptdF  31002  acunirnmpt2f  31007  funcnv4mpt  31015  cbvesum  32019  esumpfinvalf  32053  binomcxplemdvbinom  41978  binomcxplemdvsum  41980  binomcxplemnotnn0  41981  supxrleubrnmptf  42998  fnlimfv  43211  fnlimfvre2  43225  fnlimf  43226  limsupequzmptf  43279  sge0iunmptlemre  43960  smflim  44322  smflim2  44350  smfsup  44358  smfinf  44362  smflimsuplem2  44365  smflimsuplem5  44368  smflimsup  44372  smfliminf  44375