Theorem cbvmpt2x 6733

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6734 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 1843	. . . . 5 |- F/ z x e. A
2		cbvmpt2x.1	. . . . . 6 |- F/_ z B
3	2	nfcri 2758	. . . . 5 |- F/ z y e. B
4	1, 3	nfan 1828	. . . 4 |- F/ z ( x e. A /\ y e. B )
5		cbvmpt2x.3	. . . . 5 |- F/_ z C
6	5	nfeq2 2780	. . . 4 |- F/ z u = C
7	4, 6	nfan 1828	. . 3 |- F/ z ( ( x e. A /\ y e. B ) /\ u = C )
8		nfv 1843	. . . . 5 |- F/ w x e. A
9		nfcv 2764	. . . . . 6 |- F/_ w B
10	9	nfcri 2758	. . . . 5 |- F/ w y e. B
11	8, 10	nfan 1828	. . . 4 |- F/ w ( x e. A /\ y e. B )
12		cbvmpt2x.4	. . . . 5 |- F/_ w C
13	12	nfeq2 2780	. . . 4 |- F/ w u = C
14	11, 13	nfan 1828	. . 3 |- F/ w ( ( x e. A /\ y e. B ) /\ u = C )
15		nfv 1843	. . . . 5 |- F/ x z e. A
16		cbvmpt2x.2	. . . . . 6 |- F/_ x D
17	16	nfcri 2758	. . . . 5 |- F/ x w e. D
18	15, 17	nfan 1828	. . . 4 |- F/ x ( z e. A /\ w e. D )
19		cbvmpt2x.5	. . . . 5 |- F/_ x E
20	19	nfeq2 2780	. . . 4 |- F/ x u = E
21	18, 20	nfan 1828	. . 3 |- F/ x ( ( z e. A /\ w e. D ) /\ u = E )
22		nfv 1843	. . . 4 |- F/ y ( z e. A /\ w e. D )
23		cbvmpt2x.6	. . . . 5 |- F/_ y E
24	23	nfeq2 2780	. . . 4 |- F/ y u = E
25	22, 24	nfan 1828	. . 3 |- F/ y ( ( z e. A /\ w e. D ) /\ u = E )
26		eleq1 2689	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. A ) )
27	26	adantr 481	. . . . 5 |- ( ( x = z /\ y = w ) -> ( x e. A <-> z e. A ) )
28		cbvmpt2x.7	. . . . . . 7 |- ( x = z -> B = D )
29	28	eleq2d 2687	. . . . . 6 |- ( x = z -> ( y e. B <-> y e. D ) )
30		eleq1 2689	. . . . . 6 |- ( y = w -> ( y e. D <-> w e. D ) )
31	29, 30	sylan9bb 736	. . . . 5 |- ( ( x = z /\ y = w ) -> ( y e. B <-> w e. D ) )
32	27, 31	anbi12d 747	. . . 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 2632	. . . 4 |- ( ( x = z /\ y = w ) -> ( u = C <-> u = E ) )
35	32, 34	anbi12d 747	. . 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 6729	. 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 6655	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , u >. | ( ( x e. A /\ y e. B ) /\ u = C ) }
38		df-mpt2 6655	. 2 |- ( z e. A , w e. D |-> E ) = { <. <. z , w >. , u >. | ( ( z e. A /\ w e. D ) /\ u = E ) }
39	36, 37, 38	3eqtr4i 2654	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , w e. D |-> E )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   {coprab 6651   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  cbvmpt2  6734  mpt2mptsx  7233  dmmpt2ssx  7235  gsumcom2  18374  ptcmpg  21861