Theorem cbvmpt 4228

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 1581	. . . 4 |- F/ w ( x e. A /\ z = B )
2		nfv 1581	. . . . 5 |- F/ x w e. A
3		nfs1v 1999	. . . . 5 |- F/ x [ w / x ] z = B
4	2, 3	nfan 1618	. . . 4 |- F/ x ( w e. A /\ [ w / x ] z = B )
5		eleq1 2301	. . . . 5 |- ( x = w -> ( x e. A <-> w e. A ) )
6		sbequ12 1824	. . . . 5 |- ( x = w -> ( z = B <-> [ w / x ] z = B ) )
7	5, 6	anbi12d 477	. . . 4 |- ( x = w -> ( ( x e. A /\ z = B ) <-> ( w e. A /\ [ w / x ] z = B ) ) )
8	1, 4, 7	cbvopab1 4205	. . 3 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) }
9		nfv 1581	. . . . 5 |- F/ y w e. A
10		cbvmpt.1	. . . . . . 7 |- F/_ y B
11	10	nfeq2 2404	. . . . . 6 |- F/ y z = B
12	11	nfsb 2006	. . . . 5 |- F/ y [ w / x ] z = B
13	9, 12	nfan 1618	. . . 4 |- F/ y ( w e. A /\ [ w / x ] z = B )
14		nfv 1581	. . . 4 |- F/ w ( y e. A /\ z = C )
15		eleq1 2301	. . . . 5 |- ( w = y -> ( w e. A <-> y e. A ) )
16		sbequ 1893	. . . . . 6 |- ( w = y -> ( [ w / x ] z = B <-> [ y / x ] z = B ) )
17		cbvmpt.2	. . . . . . . 8 |- F/_ x C
18	17	nfeq2 2404	. . . . . . 7 |- F/ x z = C
19		cbvmpt.3	. . . . . . . 8 |- ( x = y -> B = C )
20	19	eqeq2d 2250	. . . . . . 7 |- ( x = y -> ( z = B <-> z = C ) )
21	18, 20	sbie 1844	. . . . . 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 477	. . . 4 |- ( w = y -> ( ( w e. A /\ [ w / x ] z = B ) <-> ( y e. A /\ z = C ) ) )
24	13, 14, 23	cbvopab1 4205	. . 3 |- { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
25	8, 24	eqtri 2259	. 2 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
26		df-mpt 4195	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
27		df-mpt 4195	. 2 |- ( y e. A |-> C ) = { <. y , z >. | ( y e. A /\ z = C ) }
28	25, 26, 27	3eqtr4i 2269	1 |- ( x e. A |-> B ) = ( y e. A |-> C )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   [wsb 1815   e. wcel 2209   F/_wnfc 2379   {copab 4192   |-> cmpt 4193

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-opab 4194  df-mpt 4195

This theorem is used by:  cbvmptv  4229  dffn5imf  5761  fvmpts  5786  fvmpt2  5792  mptfvex  5794  fmptcof  5878  fmptcos  5879  fliftfuns  6007  offval2  6322  qliftfuns  6897  cc2  7639  summodclem2a  12173  zsumdc  12176  fsum3cvg2  12186  cbvprod  12350  zproddc  12371  fprodseq  12375  pcmptdvds  13153  gzsumconstf  14234  gsumsnfdcmn  14257  gsum2d2fi  14274  psrass1filem  15177  cnmpt1t  15511  fsumcncntop  15793  limcmpted  15889  dvmptfsum  15951