Metamath Proof Explorer

Theorem hsmexlem3

Description: Lemma for hsmex . Clear I hypothesis and extend previous result by dominance. Note that this could be substantially strengthened, e.g., using the weak Hartogs function, but all we need here is that there be *some* dominating ordinal. (Contributed by Stefan O'Rear, 14-Feb-2015) (Revised by Mario Carneiro, 26-Jun-2015)

Ref Expression
Hypotheses hsmexlem.f 
|- F = OrdIso ( _E , B )
hsmexlem.g 
|- G = OrdIso ( _E , U_ a e. A B )
Assertion hsmexlem3 
|- ( ( ( A ~<_* D /\ C e. On ) /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> dom G e. ( har ` ~P ( D X. C ) ) )

Proof

Step Hyp Ref Expression
1 hsmexlem.f 
 |-  F = OrdIso ( _E , B )
2 hsmexlem.g 
 |-  G = OrdIso ( _E , U_ a e. A B )
3 wdomref 
 |-  ( C e. On -> C ~<_* C )
4 xpwdomg 
 |-  ( ( A ~<_* D /\ C ~<_* C ) -> ( A X. C ) ~<_* ( D X. C ) )
5 3 4 sylan2 
 |-  ( ( A ~<_* D /\ C e. On ) -> ( A X. C ) ~<_* ( D X. C ) )
6 wdompwdom 
 |-  ( ( A X. C ) ~<_* ( D X. C ) -> ~P ( A X. C ) ~<_ ~P ( D X. C ) )
7 harword 
 |-  ( ~P ( A X. C ) ~<_ ~P ( D X. C ) -> ( har ` ~P ( A X. C ) ) C_ ( har ` ~P ( D X. C ) ) )
8 5 6 7 3syl 
 |-  ( ( A ~<_* D /\ C e. On ) -> ( har ` ~P ( A X. C ) ) C_ ( har ` ~P ( D X. C ) ) )
9 8 adantr 
 |-  ( ( ( A ~<_* D /\ C e. On ) /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> ( har ` ~P ( A X. C ) ) C_ ( har ` ~P ( D X. C ) ) )
10 relwdom 
 |-  Rel ~<_*
11 10 brrelex1i 
 |-  ( A ~<_* D -> A e. _V )
12 11 adantr 
 |-  ( ( A ~<_* D /\ C e. On ) -> A e. _V )
13 12 adantr 
 |-  ( ( ( A ~<_* D /\ C e. On ) /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> A e. _V )
14 simplr 
 |-  ( ( ( A ~<_* D /\ C e. On ) /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> C e. On )
15 simpr 
 |-  ( ( ( A ~<_* D /\ C e. On ) /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> A. a e. A ( B e. ~P On /\ dom F e. C ) )
16 1 2 hsmexlem2 
 |-  ( ( A e. _V /\ C e. On /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> dom G e. ( har ` ~P ( A X. C ) ) )
17 13 14 15 16 syl3anc 
 |-  ( ( ( A ~<_* D /\ C e. On ) /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> dom G e. ( har ` ~P ( A X. C ) ) )
18 9 17 sseldd 
 |-  ( ( ( A ~<_* D /\ C e. On ) /\ A. a e. A ( B e. ~P On /\ dom F e. C ) ) -> dom G e. ( har ` ~P ( D X. C ) ) )