Theorem cbvmpt 4182

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	|- F/_ y B
cbvmpt.2	|- F/_ x C
cbvmpt.3	|- ( x = y -> B = C )

Assertion

Ref	Expression
cbvmpt	|- ( x e. A |-> B ) = ( y e. 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 |- F/ w ( x e. A /\ z = B )
2		nfv 1574	. . . . 5 |- F/ x w e. A
3		nfs1v 1990	. . . . 5 |- F/ x [ w / x ] z = B
4	2, 3	nfan 1611	. . . 4 |- F/ x ( w e. A /\ [ w / x ] z = B )
5		eleq1 2292	. . . . 5 |- ( x = w -> ( x e. A <-> w e. A ) )
6		sbequ12 1817	. . . . 5 |- ( x = w -> ( z = B <-> [ w / x ] z = B ) )
7	5, 6	anbi12d 473	. . . 4 |- ( x = w -> ( ( x e. A /\ z = B ) <-> ( w e. A /\ [ w / x ] z = B ) ) )
8	1, 4, 7	cbvopab1 4160	. . 3 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) }
9		nfv 1574	. . . . 5 |- F/ y w e. A
10		cbvmpt.1	. . . . . . 7 |- F/_ y B
11	10	nfeq2 2384	. . . . . 6 |- F/ y z = B
12	11	nfsb 1997	. . . . 5 |- F/ y [ w / x ] z = B
13	9, 12	nfan 1611	. . . 4 |- F/ y ( w e. A /\ [ w / x ] z = B )
14		nfv 1574	. . . 4 |- F/ w ( y e. A /\ z = C )
15		eleq1 2292	. . . . 5 |- ( w = y -> ( w e. A <-> y e. A ) )
16		sbequ 1886	. . . . . 6 |- ( w = y -> ( [ w / x ] z = B <-> [ y / x ] z = B ) )
17		cbvmpt.2	. . . . . . . 8 |- F/_ x C
18	17	nfeq2 2384	. . . . . . 7 |- F/ x 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 e. A /\ [ w / x ] z = B ) <-> ( y e. A /\ z = C ) ) )
24	13, 14, 23	cbvopab1 4160	. . 3 |- { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
25	8, 24	eqtri 2250	. 2 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
26		df-mpt 4150	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
27		df-mpt 4150	. 2 |- ( y e. A |-> C ) = { <. y , z >. | ( y e. A /\ z = C ) }
28	25, 26, 27	3eqtr4i 2260	1 |- ( x e. A |-> B ) = ( y e. A |-> C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   [wsb 1808   e. wcel 2200   F/_wnfc 2359   {copab 4147   |-> cmpt 4148

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 2802  df-un 3202  df-sn 3673  df-pr 3674  df-op 3676  df-opab 4149  df-mpt 4150

This theorem is referenced by:  cbvmptv  4183  dffn5imf  5697  fvmpts  5720  fvmpt2  5726  mptfvex  5728  fmptcof  5810  fmptcos  5811  fliftfuns  5934  offval2  6246  qliftfuns  6783  cc2  7476  summodclem2a  11932  zsumdc  11935  fsum3cvg2  11945  cbvprod  12109  zproddc  12130  fprodseq  12134  pcmptdvds  12908  gsumfzconstf  13919  cnmpt1t  14999  fsumcncntop  15281  limcmpted  15377  dvmptfsum  15439