Theorem cbvmptf 5185

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 2376. See cbvmptfg 5186 for a less restrictive version requiring more axioms. (Revised by GG, 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 1916	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)
2		cbvmptf.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2890	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
4		nfs1v 2162	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝑧 = 𝐵
5	3, 4	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
6		eleq1w 2819	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
7		sbequ12 2259	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
8	6, 7	anbi12d 633	. . . 4 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
9	1, 5, 8	cbvopab1 5159	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
10		cbvmptf.2	. . . . . 6 ⊢ Ⅎ𝑦𝐴
11	10	nfcri 2890	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
12		cbvmptf.3	. . . . . . 7 ⊢ Ⅎ𝑦𝐵
13	12	nfeq2 2916	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 = 𝐵
14	13	nfsbv 2335	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑧 = 𝐵
15	11, 14	nfan 1901	. . . 4 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
16		nfv 1916	. . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)
17		eleq1w 2819	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
18		cbvmptf.4	. . . . . . 7 ⊢ Ⅎ𝑥𝐶
19	18	nfeq2 2916	. . . . . 6 ⊢ Ⅎ𝑥 𝑧 = 𝐶
20		cbvmptf.5	. . . . . . 7 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
21	20	eqeq2d 2747	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
22	19, 21	sbhypf 3490	. . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
23	17, 22	anbi12d 633	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)))
24	15, 16, 23	cbvopab1 5159	. . 3 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
25	9, 24	eqtri 2759	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
26		df-mpt 5167	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
27		df-mpt 5167	. 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
28	25, 26, 27	3eqtr4i 2769	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  [wsb 2068   ∈ wcel 2114  Ⅎwnfc 2883  {copab 5147   ↦ cmpt 5166

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5148  df-mpt 5167

This theorem is referenced by:  cbvmpt  5187  resmptf  6004  fvmpt2f  6948  offval2f  7646  suppss2f  32711  fmptdF  32729  acunirnmpt2f  32734  funcnv4mpt  32741  cbvesum  34186  esumpfinvalf  34220  binomcxplemdvbinom  44780  binomcxplemdvsum  44782  binomcxplemnotnn0  44783  fmptff  45698  supxrleubrnmptf  45879  fnlimfv  46091  fnlimfvre2  46105  fnlimf  46106  limsupequzmptf  46159  sge0iunmptlemre  46843  smflim  47205  smflim2  47234  smfsup  47242  smfinf  47246  smflimsuplem2  47249  smflimsuplem5  47252  smflimsup  47256  smfliminf  47259