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Metamath Proof Explorer

Theorem elmptrab2

Description: Membership in a one-parameter class of sets, indexed by arbitrary base sets. (Contributed by Stefan O'Rear, 28-Jul-2015) (Revised by AV, 26-Mar-2021)

Ref Expression
Hypotheses elmptrab2.f 
|- F = ( x e. _V |-> { y e. B | ph } )
elmptrab2.s1 
|- ( ( x = X /\ y = Y ) -> ( ph <-> ps ) )
elmptrab2.s2 
|- ( x = X -> B = C )
elmptrab2.ex 
|- B e. _V
elmptrab2.rc 
|- ( Y e. C -> X e. W )
Assertion elmptrab2 
|- ( Y e. ( F ` X ) <-> ( Y e. C /\ ps ) )

Proof

Step Hyp Ref Expression
1 elmptrab2.f 
 |-  F = ( x e. _V |-> { y e. B | ph } )
2 elmptrab2.s1 
 |-  ( ( x = X /\ y = Y ) -> ( ph <-> ps ) )
3 elmptrab2.s2 
 |-  ( x = X -> B = C )
4 elmptrab2.ex 
 |-  B e. _V
5 elmptrab2.rc 
 |-  ( Y e. C -> X e. W )
6 4 a1i 
 |-  ( x e. _V -> B e. _V )
7 1 2 3 6 elmptrab 
 |-  ( Y e. ( F ` X ) <-> ( X e. _V /\ Y e. C /\ ps ) )
8 3simpc 
 |-  ( ( X e. _V /\ Y e. C /\ ps ) -> ( Y e. C /\ ps ) )
9 5 elexd 
 |-  ( Y e. C -> X e. _V )
10 9 adantr 
 |-  ( ( Y e. C /\ ps ) -> X e. _V )
11 simpl 
 |-  ( ( Y e. C /\ ps ) -> Y e. C )
12 simpr 
 |-  ( ( Y e. C /\ ps ) -> ps )
13 10 11 12 3jca 
 |-  ( ( Y e. C /\ ps ) -> ( X e. _V /\ Y e. C /\ ps ) )
14 8 13 impbii 
 |-  ( ( X e. _V /\ Y e. C /\ ps ) <-> ( Y e. C /\ ps ) )
15 7 14 bitri 
 |-  ( Y e. ( F ` X ) <-> ( Y e. C /\ ps ) )