Theorem cbvmpox 5857

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 5858 allows B to be a function of x. (Contributed by NM, 29-Dec-2014.)

Hypotheses

Ref	Expression
cbvmpox.1	|- F/_ z B
cbvmpox.2	|- F/_ x D
cbvmpox.3	|- F/_ z C
cbvmpox.4	|- F/_ w C
cbvmpox.5	|- F/_ x E
cbvmpox.6	|- F/_ y E
cbvmpox.7	|- ( x = z -> B = D )
cbvmpox.8	|- ( ( x = z /\ y = w ) -> C = E )

Assertion

Ref	Expression
cbvmpox	|- ( x e. A , y e. B |-> C ) = ( z e. A , w e. D |-> E )

Distinct variable groups:   x, w, y, z, A   w, B   y, D

Allowed substitution hints:   B( x, y, z)   C( x, y, z, w)   D( x, z, w)   E( x, y, z, w)

Proof of Theorem cbvmpox

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1509	. . . . 5 |- F/ z x e. A
2		cbvmpox.1	. . . . . 6 |- F/_ z B
3	2	nfcri 2276	. . . . 5 |- F/ z y e. B
4	1, 3	nfan 1545	. . . 4 |- F/ z ( x e. A /\ y e. B )
5		cbvmpox.3	. . . . 5 |- F/_ z C
6	5	nfeq2 2294	. . . 4 |- F/ z u = C
7	4, 6	nfan 1545	. . 3 |- F/ z ( ( x e. A /\ y e. B ) /\ u = C )
8		nfv 1509	. . . . 5 |- F/ w x e. A
9		nfcv 2282	. . . . . 6 |- F/_ w B
10	9	nfcri 2276	. . . . 5 |- F/ w y e. B
11	8, 10	nfan 1545	. . . 4 |- F/ w ( x e. A /\ y e. B )
12		cbvmpox.4	. . . . 5 |- F/_ w C
13	12	nfeq2 2294	. . . 4 |- F/ w u = C
14	11, 13	nfan 1545	. . 3 |- F/ w ( ( x e. A /\ y e. B ) /\ u = C )
15		nfv 1509	. . . . 5 |- F/ x z e. A
16		cbvmpox.2	. . . . . 6 |- F/_ x D
17	16	nfcri 2276	. . . . 5 |- F/ x w e. D
18	15, 17	nfan 1545	. . . 4 |- F/ x ( z e. A /\ w e. D )
19		cbvmpox.5	. . . . 5 |- F/_ x E
20	19	nfeq2 2294	. . . 4 |- F/ x u = E
21	18, 20	nfan 1545	. . 3 |- F/ x ( ( z e. A /\ w e. D ) /\ u = E )
22		nfv 1509	. . . 4 |- F/ y ( z e. A /\ w e. D )
23		cbvmpox.6	. . . . 5 |- F/_ y E
24	23	nfeq2 2294	. . . 4 |- F/ y u = E
25	22, 24	nfan 1545	. . 3 |- F/ y ( ( z e. A /\ w e. D ) /\ u = E )
26		eleq1 2203	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. A ) )
27	26	adantr 274	. . . . 5 |- ( ( x = z /\ y = w ) -> ( x e. A <-> z e. A ) )
28		cbvmpox.7	. . . . . . 7 |- ( x = z -> B = D )
29	28	eleq2d 2210	. . . . . 6 |- ( x = z -> ( y e. B <-> y e. D ) )
30		eleq1 2203	. . . . . 6 |- ( y = w -> ( y e. D <-> w e. D ) )
31	29, 30	sylan9bb 458	. . . . 5 |- ( ( x = z /\ y = w ) -> ( y e. B <-> w e. D ) )
32	27, 31	anbi12d 465	. . . 4 |- ( ( x = z /\ y = w ) -> ( ( x e. A /\ y e. B ) <-> ( z e. A /\ w e. D ) ) )
33		cbvmpox.8	. . . . 5 |- ( ( x = z /\ y = w ) -> C = E )
34	33	eqeq2d 2152	. . . 4 |- ( ( x = z /\ y = w ) -> ( u = C <-> u = E ) )
35	32, 34	anbi12d 465	. . 3 |- ( ( x = z /\ y = w ) -> ( ( ( x e. A /\ y e. B ) /\ u = C ) <-> ( ( z e. A /\ w e. D ) /\ u = E ) ) )
36	7, 14, 21, 25, 35	cbvoprab12 5853	. 2 |- { <. <. x , y >. , u >. | ( ( x e. A /\ y e. B ) /\ u = C ) } = { <. <. z , w >. , u >. | ( ( z e. A /\ w e. D ) /\ u = E ) }
37		df-mpo 5787	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , u >. | ( ( x e. A /\ y e. B ) /\ u = C ) }
38		df-mpo 5787	. 2 |- ( z e. A , w e. D |-> E ) = { <. <. z , w >. , u >. | ( ( z e. A /\ w e. D ) /\ u = E ) }
39	36, 37, 38	3eqtr4i 2171	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. D |-> E )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   F/_wnfc 2269   {coprab 5783   e. cmpo 5784

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-opab 3998  df-oprab 5786  df-mpo 5787

This theorem is referenced by:  cbvmpo  5858  mpomptsx  6103  dmmpossx  6105