Theorem cbvmpt22 39277

Description: Rule to change the second bound variable in a maps-to function, using implicit substitution. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
cbvmpt22.1	|- F/_ y A
cbvmpt22.2	|- F/_ w A
cbvmpt22.3	|- F/_ w C
cbvmpt22.4	|- F/_ y E
cbvmpt22.5	|- ( y = w -> C = E )

Assertion

Ref	Expression
cbvmpt22	|- ( x e. A , y e. B |-> C ) = ( x e. A , w e. B |-> E )

Distinct variable groups:   w, B, y   x, w, y

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

Proof of Theorem cbvmpt22

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvmpt22.2	. . . . . 6 |- F/_ w A
2	1	nfcri 2758	. . . . 5 |- F/ w x e. A
3		nfcv 2764	. . . . . 6 |- F/_ w B
4	3	nfcri 2758	. . . . 5 |- F/ w y e. B
5	2, 4	nfan 1828	. . . 4 |- F/ w ( x e. A /\ y e. B )
6		cbvmpt22.3	. . . . 5 |- F/_ w C
7	6	nfeq2 2780	. . . 4 |- F/ w u = C
8	5, 7	nfan 1828	. . 3 |- F/ w ( ( x e. A /\ y e. B ) /\ u = C )
9		cbvmpt22.1	. . . . . 6 |- F/_ y A
10	9	nfcri 2758	. . . . 5 |- F/ y x e. A
11		nfv 1843	. . . . 5 |- F/ y w e. B
12	10, 11	nfan 1828	. . . 4 |- F/ y ( x e. A /\ w e. B )
13		cbvmpt22.4	. . . . 5 |- F/_ y E
14	13	nfeq2 2780	. . . 4 |- F/ y u = E
15	12, 14	nfan 1828	. . 3 |- F/ y ( ( x e. A /\ w e. B ) /\ u = E )
16		eleq1 2689	. . . . 5 |- ( y = w -> ( y e. B <-> w e. B ) )
17	16	anbi2d 740	. . . 4 |- ( y = w -> ( ( x e. A /\ y e. B ) <-> ( x e. A /\ w e. B ) ) )
18		cbvmpt22.5	. . . . 5 |- ( y = w -> C = E )
19	18	eqeq2d 2632	. . . 4 |- ( y = w -> ( u = C <-> u = E ) )
20	17, 19	anbi12d 747	. . 3 |- ( y = w -> ( ( ( x e. A /\ y e. B ) /\ u = C ) <-> ( ( x e. A /\ w e. B ) /\ u = E ) ) )
21	8, 15, 20	cbvoprab2 6728	. 2 |- { <. <. x , y >. , u >. | ( ( x e. A /\ y e. B ) /\ u = C ) } = { <. <. x , w >. , u >. | ( ( x e. A /\ w e. B ) /\ u = E ) }
22		df-mpt2 6655	. 2 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , u >. | ( ( x e. A /\ y e. B ) /\ u = C ) }
23		df-mpt2 6655	. 2 |- ( x e. A , w e. B |-> E ) = { <. <. x , w >. , u >. | ( ( x e. A /\ w e. B ) /\ u = E ) }
24	21, 22, 23	3eqtr4i 2654	1 |- ( x e. A , y e. B |-> C ) = ( x e. A , w 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

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-oprab 6654  df-mpt2 6655

This theorem is referenced by:  smflimlem4  40982