Theorem mptssid 39450

Description: The mapping operation expressed with its actual domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
mptssid.1	|- F/_ x A
mptssid.2	|- C = { x e. A | B e. _V }

Assertion

Ref	Expression
mptssid	|- ( x e. A |-> B ) = ( x e. C |-> B )

Proof of Theorem mptssid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . . . . 10 |- ( ( x e. A /\ y = B ) -> x e. A )
2		eqvisset 3211	. . . . . . . . . . 11 |- ( y = B -> B e. _V )
3	2	adantl 482	. . . . . . . . . 10 |- ( ( x e. A /\ y = B ) -> B e. _V )
4	1, 3	jca 554	. . . . . . . . 9 |- ( ( x e. A /\ y = B ) -> ( x e. A /\ B e. _V ) )
5		rabid 3116	. . . . . . . . 9 |- ( x e. { x e. A | B e. _V } <-> ( x e. A /\ B e. _V ) )
6	4, 5	sylibr 224	. . . . . . . 8 |- ( ( x e. A /\ y = B ) -> x e. { x e. A | B e. _V } )
7		mptssid.2	. . . . . . . 8 |- C = { x e. A | B e. _V }
8	6, 7	syl6eleqr 2712	. . . . . . 7 |- ( ( x e. A /\ y = B ) -> x e. C )
9		simpr 477	. . . . . . 7 |- ( ( x e. A /\ y = B ) -> y = B )
10	8, 9	jca 554	. . . . . 6 |- ( ( x e. A /\ y = B ) -> ( x e. C /\ y = B ) )
11		mptssid.1	. . . . . . . . . . 11 |- F/_ x A
12	11	ssrab2f 39300	. . . . . . . . . 10 |- { x e. A | B e. _V } C_ A
13	7, 12	eqsstri 3635	. . . . . . . . 9 |- C C_ A
14		id 22	. . . . . . . . 9 |- ( x e. C -> x e. C )
15	13, 14	sseldi 3601	. . . . . . . 8 |- ( x e. C -> x e. A )
16	15	adantr 481	. . . . . . 7 |- ( ( x e. C /\ y = B ) -> x e. A )
17		simpr 477	. . . . . . 7 |- ( ( x e. C /\ y = B ) -> y = B )
18	16, 17	jca 554	. . . . . 6 |- ( ( x e. C /\ y = B ) -> ( x e. A /\ y = B ) )
19	10, 18	impbii 199	. . . . 5 |- ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = B ) )
20	19	ax-gen 1722	. . . 4 |- A. y ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = B ) )
21	20	ax-gen 1722	. . 3 |- A. x A. y ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = B ) )
22		eqopab2b 5005	. . 3 |- ( { <. x , y >. | ( x e. A /\ y = B ) } = { <. x , y >. | ( x e. C /\ y = B ) } <-> A. x A. y ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = B ) ) )
23	21, 22	mpbir 221	. 2 |- { <. x , y >. | ( x e. A /\ y = B ) } = { <. x , y >. | ( x e. C /\ y = B ) }
24		df-mpt 4730	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
25		df-mpt 4730	. 2 |- ( x e. C |-> B ) = { <. x , y >. | ( x e. C /\ y = B ) }
26	23, 24, 25	3eqtr4i 2654	1 |- ( x e. A |-> B ) = ( x e. C |-> B )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   F/_wnfc 2751   {crab 2916   _Vcvv 3200   {copab 4712   |-> cmpt 4729

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  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-mpt 4730

This theorem is referenced by:  limsupequzmpt2  39950  liminfequzmpt2  40023