Theorem cbvmpt 4059

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 1508	. . . 4 |- F/ w ( x e. A /\ z = B )
2		nfv 1508	. . . . 5 |- F/ x w e. A
3		nfs1v 1919	. . . . 5 |- F/ x [ w / x ] z = B
4	2, 3	nfan 1545	. . . 4 |- F/ x ( w e. A /\ [ w / x ] z = B )
5		eleq1 2220	. . . . 5 |- ( x = w -> ( x e. A <-> w e. A ) )
6		sbequ12 1751	. . . . 5 |- ( x = w -> ( z = B <-> [ w / x ] z = B ) )
7	5, 6	anbi12d 465	. . . 4 |- ( x = w -> ( ( x e. A /\ z = B ) <-> ( w e. A /\ [ w / x ] z = B ) ) )
8	1, 4, 7	cbvopab1 4037	. . 3 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) }
9		nfv 1508	. . . . 5 |- F/ y w e. A
10		cbvmpt.1	. . . . . . 7 |- F/_ y B
11	10	nfeq2 2311	. . . . . 6 |- F/ y z = B
12	11	nfsb 1926	. . . . 5 |- F/ y [ w / x ] z = B
13	9, 12	nfan 1545	. . . 4 |- F/ y ( w e. A /\ [ w / x ] z = B )
14		nfv 1508	. . . 4 |- F/ w ( y e. A /\ z = C )
15		eleq1 2220	. . . . 5 |- ( w = y -> ( w e. A <-> y e. A ) )
16		sbequ 1820	. . . . . 6 |- ( w = y -> ( [ w / x ] z = B <-> [ y / x ] z = B ) )
17		cbvmpt.2	. . . . . . . 8 |- F/_ x C
18	17	nfeq2 2311	. . . . . . 7 |- F/ x z = C
19		cbvmpt.3	. . . . . . . 8 |- ( x = y -> B = C )
20	19	eqeq2d 2169	. . . . . . 7 |- ( x = y -> ( z = B <-> z = C ) )
21	18, 20	sbie 1771	. . . . . 6 |- ( [ y / x ] z = B <-> z = C )
22	16, 21	bitrdi 195	. . . . 5 |- ( w = y -> ( [ w / x ] z = B <-> z = C ) )
23	15, 22	anbi12d 465	. . . 4 |- ( w = y -> ( ( w e. A /\ [ w / x ] z = B ) <-> ( y e. A /\ z = C ) ) )
24	13, 14, 23	cbvopab1 4037	. . 3 |- { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
25	8, 24	eqtri 2178	. 2 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
26		df-mpt 4027	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
27		df-mpt 4027	. 2 |- ( y e. A |-> C ) = { <. y , z >. | ( y e. A /\ z = C ) }
28	25, 26, 27	3eqtr4i 2188	1 |- ( x e. A |-> B ) = ( y e. A |-> C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   [wsb 1742   e. wcel 2128   F/_wnfc 2286   {copab 4024   |-> cmpt 4025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-pr 3567  df-op 3569  df-opab 4026  df-mpt 4027

This theorem is referenced by:  cbvmptv  4060  dffn5imf  5520  fvmpts  5543  fvmpt2  5548  mptfvex  5550  fmptcof  5631  fmptcos  5632  fliftfuns  5743  offval2  6041  qliftfuns  6557  cc2  7170  summodclem2a  11260  zsumdc  11263  fsum3cvg2  11273  cbvprod  11437  zproddc  11458  fprodseq  11462  cnmpt1t  12645  fsumcncntop  12916  limcmpted  12992