Theorem cbvmpt 4189

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 1577	. . . 4 |- F/ w ( x e. A /\ z = B )
2		nfv 1577	. . . . 5 |- F/ x w e. A
3		nfs1v 1992	. . . . 5 |- F/ x [ w / x ] z = B
4	2, 3	nfan 1614	. . . 4 |- F/ x ( w e. A /\ [ w / x ] z = B )
5		eleq1 2294	. . . . 5 |- ( x = w -> ( x e. A <-> w e. A ) )
6		sbequ12 1819	. . . . 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 4167	. . 3 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) }
9		nfv 1577	. . . . 5 |- F/ y w e. A
10		cbvmpt.1	. . . . . . 7 |- F/_ y B
11	10	nfeq2 2387	. . . . . 6 |- F/ y z = B
12	11	nfsb 1999	. . . . 5 |- F/ y [ w / x ] z = B
13	9, 12	nfan 1614	. . . 4 |- F/ y ( w e. A /\ [ w / x ] z = B )
14		nfv 1577	. . . 4 |- F/ w ( y e. A /\ z = C )
15		eleq1 2294	. . . . 5 |- ( w = y -> ( w e. A <-> y e. A ) )
16		sbequ 1888	. . . . . 6 |- ( w = y -> ( [ w / x ] z = B <-> [ y / x ] z = B ) )
17		cbvmpt.2	. . . . . . . 8 |- F/_ x C
18	17	nfeq2 2387	. . . . . . 7 |- F/ x z = C
19		cbvmpt.3	. . . . . . . 8 |- ( x = y -> B = C )
20	19	eqeq2d 2243	. . . . . . 7 |- ( x = y -> ( z = B <-> z = C ) )
21	18, 20	sbie 1839	. . . . . 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 4167	. . 3 |- { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
25	8, 24	eqtri 2252	. 2 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
26		df-mpt 4157	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
27		df-mpt 4157	. 2 |- ( y e. A |-> C ) = { <. y , z >. | ( y e. A /\ z = C ) }
28	25, 26, 27	3eqtr4i 2262	1 |- ( x e. A |-> B ) = ( y e. A |-> C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   [wsb 1810   e. wcel 2202   F/_wnfc 2362   {copab 4154   |-> cmpt 4155

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-mpt 4157

This theorem is referenced by:  cbvmptv  4190  dffn5imf  5710  fvmpts  5733  fvmpt2  5739  mptfvex  5741  fmptcof  5822  fmptcos  5823  fliftfuns  5949  offval2  6260  qliftfuns  6831  cc2  7529  summodclem2a  12005  zsumdc  12008  fsum3cvg2  12018  cbvprod  12182  zproddc  12203  fprodseq  12207  pcmptdvds  12981  gsumfzconstf  13992  cnmpt1t  15079  fsumcncntop  15361  limcmpted  15457  dvmptfsum  15519