Theorem cbvmpt 3851

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 1421	. . . 4 |- F/ w ( x e. A /\ z = B )
2		nfv 1421	. . . . 5 |- F/ x w e. A
3		nfs1v 1815	. . . . 5 |- F/ x [ w / x ] z = B
4	2, 3	nfan 1457	. . . 4 |- F/ x ( w e. A /\ [ w / x ] z = B )
5		eleq1 2100	. . . . 5 |- ( x = w -> ( x e. A <-> w e. A ) )
6		sbequ12 1654	. . . . 5 |- ( x = w -> ( z = B <-> [ w / x ] z = B ) )
7	5, 6	anbi12d 442	. . . 4 |- ( x = w -> ( ( x e. A /\ z = B ) <-> ( w e. A /\ [ w / x ] z = B ) ) )
8	1, 4, 7	cbvopab1 3830	. . 3 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) }
9		nfv 1421	. . . . 5 |- F/ y w e. A
10		cbvmpt.1	. . . . . . 7 |- F/_ y B
11	10	nfeq2 2189	. . . . . 6 |- F/ y z = B
12	11	nfsb 1822	. . . . 5 |- F/ y [ w / x ] z = B
13	9, 12	nfan 1457	. . . 4 |- F/ y ( w e. A /\ [ w / x ] z = B )
14		nfv 1421	. . . 4 |- F/ w ( y e. A /\ z = C )
15		eleq1 2100	. . . . 5 |- ( w = y -> ( w e. A <-> y e. A ) )
16		sbequ 1721	. . . . . 6 |- ( w = y -> ( [ w / x ] z = B <-> [ y / x ] z = B ) )
17		cbvmpt.2	. . . . . . . 8 |- F/_ x C
18	17	nfeq2 2189	. . . . . . 7 |- F/ x z = C
19		cbvmpt.3	. . . . . . . 8 |- ( x = y -> B = C )
20	19	eqeq2d 2051	. . . . . . 7 |- ( x = y -> ( z = B <-> z = C ) )
21	18, 20	sbie 1674	. . . . . 6 |- ( [ y / x ] z = B <-> z = C )
22	16, 21	syl6bb 185	. . . . 5 |- ( w = y -> ( [ w / x ] z = B <-> z = C ) )
23	15, 22	anbi12d 442	. . . 4 |- ( w = y -> ( ( w e. A /\ [ w / x ] z = B ) <-> ( y e. A /\ z = C ) ) )
24	13, 14, 23	cbvopab1 3830	. . 3 |- { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
25	8, 24	eqtri 2060	. 2 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
26		df-mpt 3820	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
27		df-mpt 3820	. 2 |- ( y e. A |-> C ) = { <. y , z >. | ( y e. A /\ z = C ) }
28	25, 26, 27	3eqtr4i 2070	1 |- ( x e. A |-> B ) = ( y e. A |-> C )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   [wsb 1645   F/_wnfc 2165   {copab 3817   |-> cmpt 3818

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-mpt 3820

This theorem is referenced by:  cbvmptv  3852  dffn5imf  5228  fvmpts  5250  fvmpt2  5254  mptfvex  5256  fmptcof  5331  fmptcos  5332  fliftfuns  5438  offval2  5726  qliftfuns  6190