Theorem isf32lem1 9175

Description: Lemma for isfin3-2 9189. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Hypotheses

Ref	Expression
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

Ref	Expression
isf32lem1	|- ( ( ( A e. om /\ B e. om ) /\ ( B C_ A /\ ph ) ) -> ( F ` A ) C_ ( F ` B ) )

Distinct variable groups:   x, B   ph, x   x, A   x, F

Allowed substitution hint:   G( x)

Proof of Theorem isf32lem1

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( a = B -> ( F ` a ) = ( F ` B ) )
2	1	sseq1d 3632	. . . 4 |- ( a = B -> ( ( F ` a ) C_ ( F ` B ) <-> ( F ` B ) C_ ( F ` B ) ) )
3	2	imbi2d 330	. . 3 |- ( a = B -> ( ( ph -> ( F ` a ) C_ ( F ` B ) ) <-> ( ph -> ( F ` B ) C_ ( F ` B ) ) ) )
4		fveq2 6191	. . . . 5 |- ( a = b -> ( F ` a ) = ( F ` b ) )
5	4	sseq1d 3632	. . . 4 |- ( a = b -> ( ( F ` a ) C_ ( F ` B ) <-> ( F ` b ) C_ ( F ` B ) ) )
6	5	imbi2d 330	. . 3 |- ( a = b -> ( ( ph -> ( F ` a ) C_ ( F ` B ) ) <-> ( ph -> ( F ` b ) C_ ( F ` B ) ) ) )
7		fveq2 6191	. . . . 5 |- ( a = suc b -> ( F ` a ) = ( F ` suc b ) )
8	7	sseq1d 3632	. . . 4 |- ( a = suc b -> ( ( F ` a ) C_ ( F ` B ) <-> ( F ` suc b ) C_ ( F ` B ) ) )
9	8	imbi2d 330	. . 3 |- ( a = suc b -> ( ( ph -> ( F ` a ) C_ ( F ` B ) ) <-> ( ph -> ( F ` suc b ) C_ ( F ` B ) ) ) )
10		fveq2 6191	. . . . 5 |- ( a = A -> ( F ` a ) = ( F ` A ) )
11	10	sseq1d 3632	. . . 4 |- ( a = A -> ( ( F ` a ) C_ ( F ` B ) <-> ( F ` A ) C_ ( F ` B ) ) )
12	11	imbi2d 330	. . 3 |- ( a = A -> ( ( ph -> ( F ` a ) C_ ( F ` B ) ) <-> ( ph -> ( F ` A ) C_ ( F ` B ) ) ) )
13		ssid 3624	. . . 4 |- ( F ` B ) C_ ( F ` B )
14	13	2a1i 12	. . 3 |- ( B e. om -> ( ph -> ( F ` B ) C_ ( F ` B ) ) )
15		isf32lem.b	. . . . . . 7 |- ( ph -> A. x e. om ( F ` suc x ) C_ ( F ` x ) )
16		suceq 5790	. . . . . . . . . 10 |- ( x = b -> suc x = suc b )
17	16	fveq2d 6195	. . . . . . . . 9 |- ( x = b -> ( F ` suc x ) = ( F ` suc b ) )
18		fveq2 6191	. . . . . . . . 9 |- ( x = b -> ( F ` x ) = ( F ` b ) )
19	17, 18	sseq12d 3634	. . . . . . . 8 |- ( x = b -> ( ( F ` suc x ) C_ ( F ` x ) <-> ( F ` suc b ) C_ ( F ` b ) ) )
20	19	rspcv 3305	. . . . . . 7 |- ( b e. om -> ( A. x e. om ( F ` suc x ) C_ ( F ` x ) -> ( F ` suc b ) C_ ( F ` b ) ) )
21	15, 20	syl5 34	. . . . . 6 |- ( b e. om -> ( ph -> ( F ` suc b ) C_ ( F ` b ) ) )
22	21	ad2antrr 762	. . . . 5 |- ( ( ( b e. om /\ B e. om ) /\ B C_ b ) -> ( ph -> ( F ` suc b ) C_ ( F ` b ) ) )
23		sstr2 3610	. . . . 5 |- ( ( F ` suc b ) C_ ( F ` b ) -> ( ( F ` b ) C_ ( F ` B ) -> ( F ` suc b ) C_ ( F ` B ) ) )
24	22, 23	syl6 35	. . . 4 |- ( ( ( b e. om /\ B e. om ) /\ B C_ b ) -> ( ph -> ( ( F ` b ) C_ ( F ` B ) -> ( F ` suc b ) C_ ( F ` B ) ) ) )
25	24	a2d 29	. . 3 |- ( ( ( b e. om /\ B e. om ) /\ B C_ b ) -> ( ( ph -> ( F ` b ) C_ ( F ` B ) ) -> ( ph -> ( F ` suc b ) C_ ( F ` B ) ) ) )
26	3, 6, 9, 12, 14, 25	findsg 7093	. 2 |- ( ( ( A e. om /\ B e. om ) /\ B C_ A ) -> ( ph -> ( F ` A ) C_ ( F ` B ) ) )
27	26	impr 649	1 |- ( ( ( A e. om /\ B e. om ) /\ ( B C_ A /\ ph ) ) -> ( F ` A ) C_ ( F ` B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ~Pcpw 4158   |^|cint 4475   ran crn 5115   suc csuc 5725   -->wf 5884   ` cfv 5888   omcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fv 5896  df-om 7066

This theorem is referenced by:  isf32lem2  9176  isf32lem3  9177