Theorem cbvmptf 4022

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 1508	. . . 4 |- F/ w ( x e. A /\ z = B )
2		cbvmptf.1	. . . . . 6 |- F/_ x A
3	2	nfcri 2275	. . . . 5 |- F/ x w e. A
4		nfs1v 1912	. . . . 5 |- F/ x [ w / x ] z = B
5	3, 4	nfan 1544	. . . 4 |- F/ x ( w e. A /\ [ w / x ] z = B )
6		eleq1w 2200	. . . . 5 |- ( x = w -> ( x e. A <-> w e. A ) )
7		sbequ12 1744	. . . . 5 |- ( x = w -> ( z = B <-> [ w / x ] z = B ) )
8	6, 7	anbi12d 464	. . . 4 |- ( x = w -> ( ( x e. A /\ z = B ) <-> ( w e. A /\ [ w / x ] z = B ) ) )
9	1, 5, 8	cbvopab1 4001	. . 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 2275	. . . . 5 |- F/ y w e. A
12		cbvmptf.3	. . . . . . 7 |- F/_ y B
13	12	nfeq2 2293	. . . . . 6 |- F/ y z = B
14	13	nfsb 1919	. . . . 5 |- F/ y [ w / x ] z = B
15	11, 14	nfan 1544	. . . 4 |- F/ y ( w e. A /\ [ w / x ] z = B )
16		nfv 1508	. . . 4 |- F/ w ( y e. A /\ z = C )
17		eleq1w 2200	. . . . 5 |- ( w = y -> ( w e. A <-> y e. A ) )
18		cbvmptf.4	. . . . . . 7 |- F/_ x C
19	18	nfeq2 2293	. . . . . 6 |- F/ x z = C
20		cbvmptf.5	. . . . . . 7 |- ( x = y -> B = C )
21	20	eqeq2d 2151	. . . . . 6 |- ( x = y -> ( z = B <-> z = C ) )
22	19, 21	sbhypf 2735	. . . . 5 |- ( w = y -> ( [ w / x ] z = B <-> z = C ) )
23	17, 22	anbi12d 464	. . . 4 |- ( w = y -> ( ( w e. A /\ [ w / x ] z = B ) <-> ( y e. A /\ z = C ) ) )
24	15, 16, 23	cbvopab1 4001	. . 3 |- { <. w , z >. | ( w e. A /\ [ w / x ] z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
25	9, 24	eqtri 2160	. 2 |- { <. x , z >. | ( x e. A /\ z = B ) } = { <. y , z >. | ( y e. A /\ z = C ) }
26		df-mpt 3991	. 2 |- ( x e. A |-> B ) = { <. x , z >. | ( x e. A /\ z = B ) }
27		df-mpt 3991	. 2 |- ( y e. A |-> C ) = { <. y , z >. | ( y e. A /\ z = C ) }
28	25, 26, 27	3eqtr4i 2170	1 |- ( x e. A |-> B ) = ( y e. A |-> C )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   [wsb 1735   F/_wnfc 2268   {copab 3988   |-> cmpt 3989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-opab 3990  df-mpt 3991

This theorem is referenced by:  resmptf  4869