Theorem cbvmptfg 5212

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. Usage of this theorem is discouraged because it depends on ax-13 2404. See cbvmptf 5211 for a version with more disjoint variable conditions, but not requiring ax-13 2404. (Contributed by NM, 11-Sep-2011.) (Revised by Thierry Arnoux, 9-Mar-2017.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

Proof of Theorem cbvmptfg

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1949	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)
2		cbvmptfg.1	. . . . . 6 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2918	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
4		nfs1v 2196	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝑧 = 𝐵
5	3, 4	nfan 1934	. . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
6		eleq1w 2847	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
7		sbequ12 2290	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
8	6, 7	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
9	1, 5, 8	cbvopab1g 5186	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
10		cbvmptfg.2	. . . . . 6 ⊢ Ⅎ𝑦𝐴
11	10	nfcri 2918	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
12		cbvmptfg.3	. . . . . . 7 ⊢ Ⅎ𝑦𝐵
13	12	nfeq2 2943	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 = 𝐵
14	13	nfsb 2555	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑧 = 𝐵
15	11, 14	nfan 1934	. . . 4 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
16		nfv 1949	. . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)
17		eleq1w 2847	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
18		sbequ 2122	. . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
19		cbvmptfg.4	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
20	19	nfeq2 2943	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝐶
21		cbvmptfg.5	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
22	21	eqeq2d 2774	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
23	20, 22	sbie 2534	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶)
24	18, 23	bitrdi 290	. . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
25	17, 24	anbi12d 644	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)))
26	15, 16, 25	cbvopab1g 5186	. . 3 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
27	9, 26	eqtri 2786	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
28		df-mpt 5193	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
29		df-mpt 5193	. 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
30	27, 28, 29	3eqtr4i 2796	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  [wsb 2101   ∈ wcel 2147  Ⅎwnfc 2911  {copab 5173   ↦ cmpt 5192

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-13 2404  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-opab 5174  df-mpt 5193

This theorem is used by:  cbvmptg  5214