Theorem cbvmpt 4147

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 1552	. . . 4 |- F/ w ( x e. A /\ z = B )
2		nfv 1552	. . . . 5 |- F/ x w e. A
3		nfs1v 1968	. . . . 5 |- F/ x [ w / x ] z = B
4	2, 3	nfan 1589	. . . 4 |- F/ x ( w e. A /\ [ w / x ] z = B )
5		eleq1 2269	. . . . 5 |- ( x = w -> ( x e. A <-> w e. A ) )
6		sbequ12 1795	. . . . 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 4125	. . 3 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) }
9		nfv 1552	. . . . 5 |- F/ y w e. A
10		cbvmpt.1	. . . . . . 7 |- F/_ y B
11	10	nfeq2 2361	. . . . . 6 |- F/ y z = B
12	11	nfsb 1975	. . . . 5 |- F/ y [ w / x ] z = B
13	9, 12	nfan 1589	. . . 4 |- F/ y ( w e. A /\ [ w / x ] z = B )
14		nfv 1552	. . . 4 |- F/ w ( y e. A /\ z = C )
15		eleq1 2269	. . . . 5 |- ( w = y -> ( w e. A <-> y e. A ) )
16		sbequ 1864	. . . . . 6 |- ( w = y -> ( [ w / x ] z = B <-> [ y / x ] z = B ) )
17		cbvmpt.2	. . . . . . . 8 |- F/_ x C
18	17	nfeq2 2361	. . . . . . 7 |- F/ x z = C
19		cbvmpt.3	. . . . . . . 8 |- ( x = y -> B = C )
20	19	eqeq2d 2218	. . . . . . 7 |- ( x = y -> ( z = B <-> z = C ) )
21	18, 20	sbie 1815	. . . . . 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 4125	. . 3 |- { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
25	8, 24	eqtri 2227	. 2 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
26		df-mpt 4115	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
27		df-mpt 4115	. 2 |- ( y e. A |-> C ) = { <. y , z >. | ( y e. A /\ z = C ) }
28	25, 26, 27	3eqtr4i 2237	1 |- ( x e. A |-> B ) = ( y e. A |-> C )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   [wsb 1786   e. wcel 2177   F/_wnfc 2336   {copab 4112   |-> cmpt 4113

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-op 3647  df-opab 4114  df-mpt 4115

This theorem is referenced by:  cbvmptv  4148  dffn5imf  5647  fvmpts  5670  fvmpt2  5676  mptfvex  5678  fmptcof  5760  fmptcos  5761  fliftfuns  5880  offval2  6187  qliftfuns  6719  cc2  7399  summodclem2a  11767  zsumdc  11770  fsum3cvg2  11780  cbvprod  11944  zproddc  11965  fprodseq  11969  pcmptdvds  12743  gsumfzconstf  13753  cnmpt1t  14832  fsumcncntop  15114  limcmpted  15210  dvmptfsum  15272