Theorem cbvmptf 4220

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 Thierry Arnoux, 9-Mar-2017.)

Hypotheses

Ref	Expression
cbvmptf.1	|- F/_ x A
cbvmptf.2	|- F/_ y A
cbvmptf.3	|- F/_ y B
cbvmptf.4	|- F/_ x C
cbvmptf.5	|- ( x = y -> B = C )

Assertion

Ref	Expression
cbvmptf	|- ( x e. A |-> B ) = ( y e. A |-> C )

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y)   B( x, y)   C( x, y)

Proof of Theorem cbvmptf

Dummy variables w z 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		cbvmptf.1	. . . . . 6 |- F/_ x A
3	2	nfcri 2386	. . . . 5 |- F/ x w e. A
4		nfs1v 1999	. . . . 5 |- F/ x [ w / x ] z = B
5	3, 4	nfan 1618	. . . 4 |- F/ x ( w e. A /\ [ w / x ] z = B )
6		eleq1w 2299	. . . . 5 |- ( x = w -> ( x e. A <-> w e. A ) )
7		sbequ12 1824	. . . . 5 |- ( x = w -> ( z = B <-> [ w / x ] z = B ) )
8	6, 7	anbi12d 477	. . . 4 |- ( x = w -> ( ( x e. A /\ z = B ) <-> ( w e. A /\ [ w / x ] z = B ) ) )
9	1, 5, 8	cbvopab1 4199	. . 3 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) }
10		cbvmptf.2	. . . . . 6 |- F/_ y A
11	10	nfcri 2386	. . . . 5 |- F/ y w e. A
12		cbvmptf.3	. . . . . . 7 |- F/_ y B
13	12	nfeq2 2404	. . . . . 6 |- F/ y z = B
14	13	nfsb 2006	. . . . 5 |- F/ y [ w / x ] z = B
15	11, 14	nfan 1618	. . . 4 |- F/ y ( w e. A /\ [ w / x ] z = B )
16		nfv 1581	. . . 4 |- F/ w ( y e. A /\ z = C )
17		eleq1w 2299	. . . . 5 |- ( w = y -> ( w e. A <-> y e. A ) )
18		cbvmptf.4	. . . . . . 7 |- F/_ x C
19	18	nfeq2 2404	. . . . . 6 |- F/ x z = C
20		cbvmptf.5	. . . . . . 7 |- ( x = y -> B = C )
21	20	eqeq2d 2250	. . . . . 6 |- ( x = y -> ( z = B <-> z = C ) )
22	19, 21	sbhypf 2872	. . . . 5 |- ( w = y -> ( [ w / x ] z = B <-> z = C ) )
23	17, 22	anbi12d 477	. . . 4 |- ( w = y -> ( ( w e. A /\ [ w / x ] z = B ) <-> ( y e. A /\ z = C ) ) )
24	15, 16, 23	cbvopab1 4199	. . 3 |- { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
25	9, 24	eqtri 2259	. 2 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
26		df-mpt 4189	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
27		df-mpt 4189	. 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 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   [wsb 1815   e. wcel 2209   F/_wnfc 2379   {copab 4186   |-> cmpt 4187

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 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 theorem 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 3711  df-pr 3712  df-op 3714  df-opab 4188  df-mpt 4189

This theorem is referenced by:  resmptf  5108