Theorem cbvmpt21 39278

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

Hypotheses

Ref	Expression
cbvmpt21.1	|- F/_ x B
cbvmpt21.2	|- F/_ z B
cbvmpt21.3	|- F/_ z C
cbvmpt21.4	|- F/_ x E
cbvmpt21.5	|- ( x = z -> C = E )

Assertion

Ref	Expression
cbvmpt21	|- ( x e. A , y e. B |-> C ) = ( z e. A , y e. B |-> E )

Distinct variable groups:   x, A, z   x, y, z

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

Proof of Theorem cbvmpt21

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . . 5 |- F/ z x e. A
2		cbvmpt21.2	. . . . . 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		cbvmpt21.3	. . . . 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/ x z e. A
9		nfcv 2764	. . . . . 6 |- F/_ x y
10		cbvmpt21.1	. . . . . 6 |- F/_ x B
11	9, 10	nfel 2777	. . . . 5 |- F/ x y e. B
12	8, 11	nfan 1828	. . . 4 |- F/ x ( z e. A /\ y e. B )
13		cbvmpt21.4	. . . . 5 |- F/_ x E
14	13	nfeq2 2780	. . . 4 |- F/ x u = E
15	12, 14	nfan 1828	. . 3 |- F/ x ( ( z e. A /\ y e. B ) /\ u = E )
16		eleq1 2689	. . . . 5 |- ( x = z -> ( x e. A <-> z e. A ) )
17	16	anbi1d 741	. . . 4 |- ( x = z -> ( ( x e. A /\ y e. B ) <-> ( z e. A /\ y e. B ) ) )
18		cbvmpt21.5	. . . . 5 |- ( x = z -> C = E )
19	18	eqeq2d 2632	. . . 4 |- ( x = z -> ( u = C <-> u = E ) )
20	17, 19	anbi12d 747	. . 3 |- ( x = z -> ( ( ( x e. A /\ y e. B ) /\ u = C ) <-> ( ( z e. A /\ y e. B ) /\ u = E ) ) )
21	7, 15, 20	cbvoprab1 6727	. 2 |- { <. <. x , y >. , u >. | ( ( x e. A /\ y e. B ) /\ u = C ) } = { <. <. z , y >. , u >. | ( ( z e. A /\ y 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 |- ( z e. A , y e. B |-> E ) = { <. <. z , y >. , u >. | ( ( z e. A /\ y e. B ) /\ u = E ) }
24	21, 22, 23	3eqtr4i 2654	1 |- ( x e. A , y e. B |-> C ) = ( z e. A , y 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: (None)