Metamath Proof Explorer

Theorem isf32lem1

Description: Lemma for isfin3-2 . Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014)

Ref Expression
Hypotheses isf32lem.a 
|- ( ph -> F : _om --> ~P G )
isf32lem.b 
|- ( ph -> A. x e. _om ( F ` suc x ) C_ ( F ` x ) )
isf32lem.c 
|- ( ph -> -. |^| ran F e. ran F )
Assertion isf32lem1 
|- ( ( ( A e. _om /\ B e. _om ) /\ ( B C_ A /\ ph ) ) -> ( F ` A ) C_ ( F ` B ) )

Proof

Step Hyp Ref Expression
1 isf32lem.a 
 |-  ( ph -> F : _om --> ~P G )
2 isf32lem.b 
 |-  ( ph -> A. x e. _om ( F ` suc x ) C_ ( F ` x ) )
3 isf32lem.c 
 |-  ( ph -> -. |^| ran F e. ran F )
4 fveq2 
 |-  ( a = B -> ( F ` a ) = ( F ` B ) )
5 4 sseq1d 
 |-  ( a = B -> ( ( F ` a ) C_ ( F ` B ) <-> ( F ` B ) C_ ( F ` B ) ) )
6 5 imbi2d 
 |-  ( a = B -> ( ( ph -> ( F ` a ) C_ ( F ` B ) ) <-> ( ph -> ( F ` B ) C_ ( F ` B ) ) ) )
7 fveq2 
 |-  ( a = b -> ( F ` a ) = ( F ` b ) )
8 7 sseq1d 
 |-  ( a = b -> ( ( F ` a ) C_ ( F ` B ) <-> ( F ` b ) C_ ( F ` B ) ) )
9 8 imbi2d 
 |-  ( a = b -> ( ( ph -> ( F ` a ) C_ ( F ` B ) ) <-> ( ph -> ( F ` b ) C_ ( F ` B ) ) ) )
10 fveq2 
 |-  ( a = suc b -> ( F ` a ) = ( F ` suc b ) )
11 10 sseq1d 
 |-  ( a = suc b -> ( ( F ` a ) C_ ( F ` B ) <-> ( F ` suc b ) C_ ( F ` B ) ) )
12 11 imbi2d 
 |-  ( a = suc b -> ( ( ph -> ( F ` a ) C_ ( F ` B ) ) <-> ( ph -> ( F ` suc b ) C_ ( F ` B ) ) ) )
13 fveq2 
 |-  ( a = A -> ( F ` a ) = ( F ` A ) )
14 13 sseq1d 
 |-  ( a = A -> ( ( F ` a ) C_ ( F ` B ) <-> ( F ` A ) C_ ( F ` B ) ) )
15 14 imbi2d 
 |-  ( a = A -> ( ( ph -> ( F ` a ) C_ ( F ` B ) ) <-> ( ph -> ( F ` A ) C_ ( F ` B ) ) ) )
16 ssid 
 |-  ( F ` B ) C_ ( F ` B )
17 16 2a1i 
 |-  ( B e. _om -> ( ph -> ( F ` B ) C_ ( F ` B ) ) )
18 suceq 
 |-  ( x = b -> suc x = suc b )
19 18 fveq2d 
 |-  ( x = b -> ( F ` suc x ) = ( F ` suc b ) )
20 fveq2 
 |-  ( x = b -> ( F ` x ) = ( F ` b ) )
21 19 20 sseq12d 
 |-  ( x = b -> ( ( F ` suc x ) C_ ( F ` x ) <-> ( F ` suc b ) C_ ( F ` b ) ) )
22 21 rspcv 
 |-  ( b e. _om -> ( A. x e. _om ( F ` suc x ) C_ ( F ` x ) -> ( F ` suc b ) C_ ( F ` b ) ) )
23 2 22 syl5 
 |-  ( b e. _om -> ( ph -> ( F ` suc b ) C_ ( F ` b ) ) )
24 23 ad2antrr 
 |-  ( ( ( b e. _om /\ B e. _om ) /\ B C_ b ) -> ( ph -> ( F ` suc b ) C_ ( F ` b ) ) )
25 sstr2 
 |-  ( ( F ` suc b ) C_ ( F ` b ) -> ( ( F ` b ) C_ ( F ` B ) -> ( F ` suc b ) C_ ( F ` B ) ) )
26 24 25 syl6 
 |-  ( ( ( b e. _om /\ B e. _om ) /\ B C_ b ) -> ( ph -> ( ( F ` b ) C_ ( F ` B ) -> ( F ` suc b ) C_ ( F ` B ) ) ) )
27 26 a2d 
 |-  ( ( ( b e. _om /\ B e. _om ) /\ B C_ b ) -> ( ( ph -> ( F ` b ) C_ ( F ` B ) ) -> ( ph -> ( F ` suc b ) C_ ( F ` B ) ) ) )
28 6 9 12 15 17 27 findsg 
 |-  ( ( ( A e. _om /\ B e. _om ) /\ B C_ A ) -> ( ph -> ( F ` A ) C_ ( F ` B ) ) )
29 28 impr 
 |-  ( ( ( A e. _om /\ B e. _om ) /\ ( B C_ A /\ ph ) ) -> ( F ` A ) C_ ( F ` B ) )