Theorem cbvmpt2x2 42114

Description: Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6734 allows A to be a function of y, analogous to cbvmpt2x 6733. (Contributed by AV, 30-Mar-2019.)

Hypotheses

Ref	Expression
cbvmpt2x2.1	|- F/_ z A
cbvmpt2x2.2	|- F/_ y D
cbvmpt2x2.3	|- F/_ z C
cbvmpt2x2.4	|- F/_ w C
cbvmpt2x2.5	|- F/_ x E
cbvmpt2x2.6	|- F/_ y E
cbvmpt2x2.7	|- ( y = z -> A = D )
cbvmpt2x2.8	|- ( ( y = z /\ x = w ) -> C = E )

Assertion

Ref	Expression
cbvmpt2x2	|- ( x e. A , y e. B |-> C ) = ( w e. D , z e. B |-> E )

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

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

Proof of Theorem cbvmpt2x2

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . 5 |- F/ w x e. A
2		nfv 1843	. . . . 5 |- F/ w y e. B
3	1, 2	nfan 1828	. . . 4 |- F/ w ( x e. A /\ y e. B )
4		cbvmpt2x2.4	. . . . 5 |- F/_ w C
5	4	nfeq2 2780	. . . 4 |- F/ w u = C
6	3, 5	nfan 1828	. . 3 |- F/ w ( ( x e. A /\ y e. B ) /\ u = C )
7		cbvmpt2x2.1	. . . . . 6 |- F/_ z A
8	7	nfcri 2758	. . . . 5 |- F/ z x e. A
9		nfv 1843	. . . . 5 |- F/ z y e. B
10	8, 9	nfan 1828	. . . 4 |- F/ z ( x e. A /\ y e. B )
11		cbvmpt2x2.3	. . . . 5 |- F/_ z C
12	11	nfeq2 2780	. . . 4 |- F/ z u = C
13	10, 12	nfan 1828	. . 3 |- F/ z ( ( x e. A /\ y e. B ) /\ u = C )
14		nfv 1843	. . . . 5 |- F/ x w e. D
15		nfv 1843	. . . . 5 |- F/ x z e. B
16	14, 15	nfan 1828	. . . 4 |- F/ x ( w e. D /\ z e. B )
17		cbvmpt2x2.5	. . . . 5 |- F/_ x E
18	17	nfeq2 2780	. . . 4 |- F/ x u = E
19	16, 18	nfan 1828	. . 3 |- F/ x ( ( w e. D /\ z e. B ) /\ u = E )
20		cbvmpt2x2.2	. . . . . 6 |- F/_ y D
21	20	nfcri 2758	. . . . 5 |- F/ y w e. D
22		nfv 1843	. . . . 5 |- F/ y z e. B
23	21, 22	nfan 1828	. . . 4 |- F/ y ( w e. D /\ z e. B )
24		cbvmpt2x2.6	. . . . 5 |- F/_ y E
25	24	nfeq2 2780	. . . 4 |- F/ y u = E
26	23, 25	nfan 1828	. . 3 |- F/ y ( ( w e. D /\ z e. B ) /\ u = E )
27		eleq1 2689	. . . . . 6 |- ( x = w -> ( x e. A <-> w e. A ) )
28		cbvmpt2x2.7	. . . . . . 7 |- ( y = z -> A = D )
29	28	eleq2d 2687	. . . . . 6 |- ( y = z -> ( w e. A <-> w e. D ) )
30	27, 29	sylan9bb 736	. . . . 5 |- ( ( x = w /\ y = z ) -> ( x e. A <-> w e. D ) )
31		simpr 477	. . . . . 6 |- ( ( x = w /\ y = z ) -> y = z )
32	31	eleq1d 2686	. . . . 5 |- ( ( x = w /\ y = z ) -> ( y e. B <-> z e. B ) )
33	30, 32	anbi12d 747	. . . 4 |- ( ( x = w /\ y = z ) -> ( ( x e. A /\ y e. B ) <-> ( w e. D /\ z e. B ) ) )
34		cbvmpt2x2.8	. . . . . 6 |- ( ( y = z /\ x = w ) -> C = E )
35	34	ancoms 469	. . . . 5 |- ( ( x = w /\ y = z ) -> C = E )
36	35	eqeq2d 2632	. . . 4 |- ( ( x = w /\ y = z ) -> ( u = C <-> u = E ) )
37	33, 36	anbi12d 747	. . 3 |- ( ( x = w /\ y = z ) -> ( ( ( x e. A /\ y e. B ) /\ u = C ) <-> ( ( w e. D /\ z e. B ) /\ u = E ) ) )
38	6, 13, 19, 26, 37	cbvoprab12 6729	. 2 |- { <. <. x , y >. , u >. | ( ( x e. A /\ y e. B ) /\ u = C ) } = { <. <. w , z >. , u >. | ( ( w e. D /\ z e. B ) /\ u = E ) }
39		df-mpt2 6655	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , u >. | ( ( x e. A /\ y e. B ) /\ u = C ) }
40		df-mpt2 6655	. 2 |- ( w e. D , z e. B |-> E ) = { <. <. w , z >. , u >. | ( ( w e. D /\ z e. B ) /\ u = E ) }
41	38, 39, 40	3eqtr4i 2654	1 |- ( x e. A , y e. B |-> C ) = ( w e. D , z e. B |-> E )

Syntax hints:   -> wi 4   /\ 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:  dmmpt2ssx2  42115