Theorem cbvmpt 4179

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 1574	. . . 4 ⊢ Ⅎ𝑤(𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)
2		nfv 1574	. . . . 5 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐴
3		nfs1v 1990	. . . . 5 ⊢ Ⅎ𝑥[𝑤 / 𝑥]𝑧 = 𝐵
4	2, 3	nfan 1611	. . . 4 ⊢ Ⅎ𝑥(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
5		eleq1 2292	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴))
6		sbequ12 1817	. . . . 5 ⊢ (𝑥 = 𝑤 → (𝑧 = 𝐵 ↔ [𝑤 / 𝑥]𝑧 = 𝐵))
7	5, 6	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝑤 → ((𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)))
8	1, 4, 7	cbvopab1 4157	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)}
9		nfv 1574	. . . . 5 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐴
10		cbvmpt.1	. . . . . . 7 ⊢ Ⅎ𝑦𝐵
11	10	nfeq2 2384	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 = 𝐵
12	11	nfsb 1997	. . . . 5 ⊢ Ⅎ𝑦[𝑤 / 𝑥]𝑧 = 𝐵
13	9, 12	nfan 1611	. . . 4 ⊢ Ⅎ𝑦(𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)
14		nfv 1574	. . . 4 ⊢ Ⅎ𝑤(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)
15		eleq1 2292	. . . . 5 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
16		sbequ 1886	. . . . . 6 ⊢ (𝑤 = 𝑦 → ([𝑤 / 𝑥]𝑧 = 𝐵 ↔ [𝑦 / 𝑥]𝑧 = 𝐵))
17		cbvmpt.2	. . . . . . . 8 ⊢ Ⅎ𝑥𝐶
18	17	nfeq2 2384	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = 𝐶
19		cbvmpt.3	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
20	19	eqeq2d 2241	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑧 = 𝐵 ↔ 𝑧 = 𝐶))
21	18, 20	sbie 1837	. . . . . 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 473	. . . 4 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)))
24	13, 14, 23	cbvopab1 4157	. . 3 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝐴 ∧ [𝑤 / 𝑥]𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
25	8, 24	eqtri 2250	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
26		df-mpt 4147	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
27		df-mpt 4147	. 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐶) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐶)}
28	25, 26, 27	3eqtr4i 2260	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  [wsb 1808   ∈ wcel 2200  Ⅎwnfc 2359  {copab 4144   ↦ cmpt 4145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146  df-mpt 4147

This theorem is referenced by:  cbvmptv  4180  dffn5imf  5694  fvmpts  5717  fvmpt2  5723  mptfvex  5725  fmptcof  5807  fmptcos  5808  fliftfuns  5931  offval2  6243  qliftfuns  6779  cc2  7469  summodclem2a  11913  zsumdc  11916  fsum3cvg2  11926  cbvprod  12090  zproddc  12111  fprodseq  12115  pcmptdvds  12889  gsumfzconstf  13900  cnmpt1t  14980  fsumcncntop  15262  limcmpted  15358  dvmptfsum  15420