Theorem cbvmptf 5219

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 2370. See cbvmptfg 5220 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 1917	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)
2		cbvmptf.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2889	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
4		nfs1v 2153	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝑧 = 𝐵
5	3, 4	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
6		eleq1w 2815	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
7		sbequ12 2243	. . . . 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 5185	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
10		cbvmptf.2	. . . . . 6 ⊢ Ⅎ𝑦𝐴
11	10	nfcri 2889	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
12		cbvmptf.3	. . . . . . 7 ⊢ Ⅎ𝑦𝐵
13	12	nfeq2 2919	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 = 𝐵
14	13	nfsbv 2323	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑧 = 𝐵
15	11, 14	nfan 1902	. . . 4 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
16		nfv 1917	. . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)
17		eleq1w 2815	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
18		sbequ 2086	. . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
19		cbvmptf.4	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
20	19	nfeq2 2919	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝐶
21		cbvmptf.5	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
22	21	eqeq2d 2742	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
23	20, 22	sbiev 2308	. . . . . 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 631	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)))
26	15, 16, 25	cbvopab1 5185	. . 3 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
27	9, 26	eqtri 2759	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
28		df-mpt 5194	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
29		df-mpt 5194	. 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
30	27, 28, 29	3eqtr4i 2769	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  [wsb 2067   ∈ wcel 2106  Ⅎwnfc 2882  {copab 5172   ↦ cmpt 5193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173  df-mpt 5194

This theorem is referenced by:  cbvmpt  5221  resmptf  5998  fvmpt2f  6954  offval2f  7637  suppss2f  31620  fmptdF  31639  acunirnmpt2f  31644  funcnv4mpt  31652  cbvesum  32730  esumpfinvalf  32764  binomcxplemdvbinom  42755  binomcxplemdvsum  42757  binomcxplemnotnn0  42758  fmptff  43619  supxrleubrnmptf  43806  fnlimfv  44024  fnlimfvre2  44038  fnlimf  44039  limsupequzmptf  44092  sge0iunmptlemre  44776  smflim  45138  smflim2  45167  smfsup  45175  smfinf  45179  smflimsuplem2  45182  smflimsuplem5  45185  smflimsup  45189  smfliminf  45192