Theorem cbvmpt 3981

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 1489	. . . 4 |- F/ w ( x e. A /\ z = B )
2		nfv 1489	. . . . 5 |- F/ x w e. A
3		nfs1v 1888	. . . . 5 |- F/ x [ w / x ] z = B
4	2, 3	nfan 1525	. . . 4 |- F/ x ( w e. A /\ [ w / x ] z = B )
5		eleq1 2175	. . . . 5 |- ( x = w -> ( x e. A <-> w e. A ) )
6		sbequ12 1725	. . . . 5 |- ( x = w -> ( z = B <-> [ w / x ] z = B ) )
7	5, 6	anbi12d 462	. . . 4 |- ( x = w -> ( ( x e. A /\ z = B ) <-> ( w e. A /\ [ w / x ] z = B ) ) )
8	1, 4, 7	cbvopab1 3959	. . 3 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) }
9		nfv 1489	. . . . 5 |- F/ y w e. A
10		cbvmpt.1	. . . . . . 7 |- F/_ y B
11	10	nfeq2 2265	. . . . . 6 |- F/ y z = B
12	11	nfsb 1895	. . . . 5 |- F/ y [ w / x ] z = B
13	9, 12	nfan 1525	. . . 4 |- F/ y ( w e. A /\ [ w / x ] z = B )
14		nfv 1489	. . . 4 |- F/ w ( y e. A /\ z = C )
15		eleq1 2175	. . . . 5 |- ( w = y -> ( w e. A <-> y e. A ) )
16		sbequ 1792	. . . . . 6 |- ( w = y -> ( [ w / x ] z = B <-> [ y / x ] z = B ) )
17		cbvmpt.2	. . . . . . . 8 |- F/_ x C
18	17	nfeq2 2265	. . . . . . 7 |- F/ x z = C
19		cbvmpt.3	. . . . . . . 8 |- ( x = y -> B = C )
20	19	eqeq2d 2124	. . . . . . 7 |- ( x = y -> ( z = B <-> z = C ) )
21	18, 20	sbie 1745	. . . . . 6 |- ( [ y / x ] z = B <-> z = C )
22	16, 21	syl6bb 195	. . . . 5 |- ( w = y -> ( [ w / x ] z = B <-> z = C ) )
23	15, 22	anbi12d 462	. . . 4 |- ( w = y -> ( ( w e. A /\ [ w / x ] z = B ) <-> ( y e. A /\ z = C ) ) )
24	13, 14, 23	cbvopab1 3959	. . 3 |- { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
25	8, 24	eqtri 2133	. 2 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
26		df-mpt 3949	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
27		df-mpt 3949	. 2 |- ( y e. A |-> C ) = { <. y , z >. | ( y e. A /\ z = C ) }
28	25, 26, 27	3eqtr4i 2143	1 |- ( x e. A |-> B ) = ( y e. A |-> C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1312   e. wcel 1461   [wsb 1716   F/_wnfc 2240   {copab 3946   |-> cmpt 3947

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-sn 3497  df-pr 3498  df-op 3500  df-opab 3948  df-mpt 3949

This theorem is referenced by:  cbvmptv  3982  dffn5imf  5428  fvmpts  5451  fvmpt2  5456  mptfvex  5458  fmptcof  5539  fmptcos  5540  fliftfuns  5651  offval2  5949  qliftfuns  6465  summodclem2a  11036  zsumdc  11039  fsum3cvg2  11049  cnmpt1t  12290  fsumcncntop  12536  limcmpted  12582