Theorem cbvmpt 4226

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.)

Hypotheses

Ref	Expression
cbvmpt.1	⊢ Ⅎ𝑦𝐵
cbvmpt.2	⊢ Ⅎ𝑥𝐶
cbvmpt.3	⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑦,𝐴

Allowed substitution hints:   𝐵(𝑥, 𝑦)   𝐶(𝑥, 𝑦)

Proof of Theorem cbvmpt

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

Step	Hyp	Ref	Expression
1		nfv 1581	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)
2		nfv 1581	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
3		nfs1v 1999	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝑧 = 𝐵
4	2, 3	nfan 1618	. . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
5		eleq1 2301	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
6		sbequ12 1824	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
7	5, 6	anbi12d 477	. . . 4 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
8	1, 4, 7	cbvopab1 4204	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
9		nfv 1581	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
10		cbvmpt.1	. . . . . . 7 ⊢ Ⅎ𝑦𝐵
11	10	nfeq2 2404	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 = 𝐵
12	11	nfsb 2006	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑧 = 𝐵
13	9, 12	nfan 1618	. . . 4 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
14		nfv 1581	. . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)
15		eleq1 2301	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
16		sbequ 1893	. . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
17		cbvmpt.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
18	17	nfeq2 2404	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝐶
19		cbvmpt.3	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
20	19	eqeq2d 2250	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
21	18, 20	sbie 1844	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶)
22	16, 21	bitrdi 196	. . . . 5 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
23	15, 22	anbi12d 477	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)))
24	13, 14, 23	cbvopab1 4204	. . 3 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
25	8, 24	eqtri 2259	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
26		df-mpt 4194	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
27		df-mpt 4194	. 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
28	25, 26, 27	3eqtr4i 2269	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402  [wsb 1815   ∈ wcel 2209  Ⅎwnfc 2379  {copab 4191   ↦ cmpt 4192

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-opab 4193  df-mpt 4194

This theorem is used by:  cbvmptv  4227  dffn5imf  5758  fvmpts  5783  fvmpt2  5789  mptfvex  5791  fmptcof  5875  fmptcos  5876  fliftfuns  6004  offval2  6318  qliftfuns  6893  cc2  7633  summodclem2a  12150  zsumdc  12153  fsum3cvg2  12163  cbvprod  12327  zproddc  12348  fprodseq  12352  pcmptdvds  13126  gzsumconstf  14146  cnmpt1t  15388  fsumcncntop  15670  limcmpted  15766  dvmptfsum  15828