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Metamath Proof Explorer

Theorem mptssid

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

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

Proof

Step Hyp Ref Expression
1 mptssid.1 
 |-  F/_ x A
2 mptssid.2 
 |-  C = { x e. A | B e. _V }
3 eqvisset 
 |-  ( y = B -> B e. _V )
4 3 anim2i 
 |-  ( ( x e. A /\ y = B ) -> ( x e. A /\ B e. _V ) )
5 rabid 
 |-  ( x e. { x e. A | B e. _V } <-> ( x e. A /\ B e. _V ) )
6 4 5 sylibr 
 |-  ( ( x e. A /\ y = B ) -> x e. { x e. A | B e. _V } )
7 6 2 eleqtrrdi 
 |-  ( ( x e. A /\ y = B ) -> x e. C )
8 simpr 
 |-  ( ( x e. A /\ y = B ) -> y = B )
9 7 8 jca 
 |-  ( ( x e. A /\ y = B ) -> ( x e. C /\ y = B ) )
10 1 ssrab2f 
 |-  { x e. A | B e. _V } C_ A
11 2 10 eqsstri 
 |-  C C_ A
12 11 sseli 
 |-  ( x e. C -> x e. A )
13 12 anim1i 
 |-  ( ( x e. C /\ y = B ) -> ( x e. A /\ y = B ) )
14 9 13 impbii 
 |-  ( ( x e. A /\ y = B ) <-> ( x e. C /\ y = B ) )
15 14 opabbii 
 |-  { <. x , y >. | ( x e. A /\ y = B ) } = { <. x , y >. | ( x e. C /\ y = B ) }
16 df-mpt 
 |-  ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
17 df-mpt 
 |-  ( x e. C |-> B ) = { <. x , y >. | ( x e. C /\ y = B ) }
18 15 16 17 3eqtr4i 
 |-  ( x e. A |-> B ) = ( x e. C |-> B )