Theorem cbvmpt 4178

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 4156	. . 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 4156	. . 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 4146	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
27		df-mpt 4146	. 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 4143   |-> cmpt 4144

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 4145  df-mpt 4146

This theorem is referenced by:  cbvmptv  4179  dffn5imf  5688  fvmpts  5711  fvmpt2  5717  mptfvex  5719  fmptcof  5801  fmptcos  5802  fliftfuns  5921  offval2  6232  qliftfuns  6764  cc2  7449  summodclem2a  11887  zsumdc  11890  fsum3cvg2  11900  cbvprod  12064  zproddc  12085  fprodseq  12089  pcmptdvds  12863  gsumfzconstf  13874  cnmpt1t  14953  fsumcncntop  15235  limcmpted  15331  dvmptfsum  15393