Theorem cbvmpt2x 5582

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 5583 allows B to be a function of x. (Contributed by NM, 29-Dec-2014.)

Hypotheses

Ref	Expression
cbvmpt2x.1	|- F/_ z B
cbvmpt2x.2	|- F/_ x D
cbvmpt2x.3	|- F/_ z C
cbvmpt2x.4	|- F/_ w C
cbvmpt2x.5	|- F/_ x E
cbvmpt2x.6	|- F/_ y E
cbvmpt2x.7	|- ( x = z -> B = D )
cbvmpt2x.8	|- ( ( x = z /\ y = w ) -> C = E )

Assertion

Ref	Expression
cbvmpt2x	|- ( 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 cbvmpt2x

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1421	. . . . 5 |- F/ z x e. A
2		cbvmpt2x.1	. . . . . 6 |- F/_ z B
3	2	nfcri 2172	. . . . 5 |- F/ z y e. B
4	1, 3	nfan 1457	. . . 4 |- F/ z ( x e. A /\ y e. B )
5		cbvmpt2x.3	. . . . 5 |- F/_ z C
6	5	nfeq2 2189	. . . 4 |- F/ z u = C
7	4, 6	nfan 1457	. . 3 |- F/ z ( ( x e. A /\ y e. B ) /\ u = C )
8		nfv 1421	. . . . 5 |- F/ w x e. A
9		nfcv 2178	. . . . . 6 |- F/_ w B
10	9	nfcri 2172	. . . . 5 |- F/ w y e. B
11	8, 10	nfan 1457	. . . 4 |- F/ w ( x e. A /\ y e. B )
12		cbvmpt2x.4	. . . . 5 |- F/_ w C
13	12	nfeq2 2189	. . . 4 |- F/ w u = C
14	11, 13	nfan 1457	. . 3 |- F/ w ( ( x e. A /\ y e. B ) /\ u = C )
15		nfv 1421	. . . . 5 |- F/ x z e. A
16		cbvmpt2x.2	. . . . . 6 |- F/_ x D
17	16	nfcri 2172	. . . . 5 |- F/ x w e. D
18	15, 17	nfan 1457	. . . 4 |- F/ x ( z e. A /\ w e. D )
19		cbvmpt2x.5	. . . . 5 |- F/_ x E
20	19	nfeq2 2189	. . . 4 |- F/ x u = E
21	18, 20	nfan 1457	. . 3 |- F/ x ( ( z e. A /\ w e. D ) /\ u = E )
22		nfv 1421	. . . 4 |- F/ y ( z e. A /\ w e. D )
23		cbvmpt2x.6	. . . . 5 |- F/_ y E
24	23	nfeq2 2189	. . . 4 |- F/ y u = E
25	22, 24	nfan 1457	. . 3 |- F/ y ( ( z e. A /\ w e. D ) /\ u = E )
26		eleq1 2100	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. A ) )
27	26	adantr 261	. . . . 5 |- ( ( x = z /\ y = w ) -> ( x e. A <-> z e. A ) )
28		cbvmpt2x.7	. . . . . . 7 |- ( x = z -> B = D )
29	28	eleq2d 2107	. . . . . 6 |- ( x = z -> ( y e. B <-> y e. D ) )
30		eleq1 2100	. . . . . 6 |- ( y = w -> ( y e. D <-> w e. D ) )
31	29, 30	sylan9bb 435	. . . . 5 |- ( ( x = z /\ y = w ) -> ( y e. B <-> w e. D ) )
32	27, 31	anbi12d 442	. . . 4 |- ( ( x = z /\ y = w ) -> ( ( x e. A /\ y e. B ) <-> ( z e. A /\ w e. D ) ) )
33		cbvmpt2x.8	. . . . 5 |- ( ( x = z /\ y = w ) -> C = E )
34	33	eqeq2d 2051	. . . 4 |- ( ( x = z /\ y = w ) -> ( u = C <-> u = E ) )
35	32, 34	anbi12d 442	. . 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 5578	. 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-mpt2 5517	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , u >. | ( ( x e. A /\ y e. B ) /\ u = C ) }
38		df-mpt2 5517	. 2 |- ( z e. A , w e. D |-> E ) = { <. <. z , w >. , u >. | ( ( z e. A /\ w e. D ) /\ u = E ) }
39	36, 37, 38	3eqtr4i 2070	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. D |-> E )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   F/_wnfc 2165   {coprab 5513   |-> cmpt2 5514

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-oprab 5516  df-mpt2 5517

This theorem is referenced by:  cbvmpt2  5583  mpt2mptsx  5823  dmmpt2ssx  5825