Metamath Proof Explorer

Theorem isowe2

Description: A weak form of isowe that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014)

Ref Expression
Assertion isowe2 
|- ( ( H Isom R , S ( A , B ) /\ A. x ( H " x ) e. _V ) -> ( S We B -> R We A ) )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( H Isom R , S ( A , B ) /\ A. x ( H " x ) e. _V ) -> H Isom R , S ( A , B ) )
2 imaeq2 
 |-  ( x = y -> ( H " x ) = ( H " y ) )
3 2 eleq1d 
 |-  ( x = y -> ( ( H " x ) e. _V <-> ( H " y ) e. _V ) )
4 3 spvv 
 |-  ( A. x ( H " x ) e. _V -> ( H " y ) e. _V )
5 4 adantl 
 |-  ( ( H Isom R , S ( A , B ) /\ A. x ( H " x ) e. _V ) -> ( H " y ) e. _V )
6 1 5 isofrlem 
 |-  ( ( H Isom R , S ( A , B ) /\ A. x ( H " x ) e. _V ) -> ( S Fr B -> R Fr A ) )
7 isosolem 
 |-  ( H Isom R , S ( A , B ) -> ( S Or B -> R Or A ) )
8 7 adantr 
 |-  ( ( H Isom R , S ( A , B ) /\ A. x ( H " x ) e. _V ) -> ( S Or B -> R Or A ) )
9 6 8 anim12d 
 |-  ( ( H Isom R , S ( A , B ) /\ A. x ( H " x ) e. _V ) -> ( ( S Fr B /\ S Or B ) -> ( R Fr A /\ R Or A ) ) )
10 df-we 
 |-  ( S We B <-> ( S Fr B /\ S Or B ) )
11 df-we 
 |-  ( R We A <-> ( R Fr A /\ R Or A ) )
12 9 10 11 3imtr4g 
 |-  ( ( H Isom R , S ( A , B ) /\ A. x ( H " x ) e. _V ) -> ( S We B -> R We A ) )