Theorem fpwwe2lem10 9461

Description: Lemma for fpwwe2 9465. Given two well-orders <. X , R >. and <. Y , S >. of parts of A, one is an initial segment of the other. (Contributed by Mario Carneiro, 15-May-2015.)

Hypotheses

Ref	Expression
fpwwe2.1	|- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }
fpwwe2.2	|- ( ph -> A e. _V )
fpwwe2.3	|- ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A )
fpwwe2lem10.4	|- ( ph -> X W R )
fpwwe2lem10.6	|- ( ph -> Y W S )

Assertion

Ref	Expression
fpwwe2lem10	|- ( ph -> ( ( X C_ Y /\ R = ( S i^i ( Y X. X ) ) ) \/ ( Y C_ X /\ S = ( R i^i ( X X. Y ) ) ) ) )

Distinct variable groups:   y, u, r, x, F   X, r, u, x, y   ph, r, u, x, y   A, r, x   R, r, u, x, y   Y, r, u, x, y   S, r, u, x, y   W, r, u, x, y

Allowed substitution hints:   A( y, u)

Proof of Theorem fpwwe2lem10

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- OrdIso ( R , X ) = OrdIso ( R , X )
2	1	oicl 8434	. . 3 |- Ord dom OrdIso ( R , X )
3		eqid 2622	. . . 4 |- OrdIso ( S , Y ) = OrdIso ( S , Y )
4	3	oicl 8434	. . 3 |- Ord dom OrdIso ( S , Y )
5		ordtri2or2 5823	. . 3 |- ( ( Ord dom OrdIso ( R , X ) /\ Ord dom OrdIso ( S , Y ) ) -> ( dom OrdIso ( R , X ) C_ dom OrdIso ( S , Y ) \/ dom OrdIso ( S , Y ) C_ dom OrdIso ( R , X ) ) )
6	2, 4, 5	mp2an 708	. 2 |- ( dom OrdIso ( R , X ) C_ dom OrdIso ( S , Y ) \/ dom OrdIso ( S , Y ) C_ dom OrdIso ( R , X ) )
7		fpwwe2.1	. . . . 5 |- W = { <. x , r >. | ( ( x C_ A /\ r C_ ( x X. x ) ) /\ ( r We x /\ A. y e. x [. ( `' r " { y } ) / u ]. ( u F ( r i^i ( u X. u ) ) ) = y ) ) }
8		fpwwe2.2	. . . . . 6 |- ( ph -> A e. _V )
9	8	adantr 481	. . . . 5 |- ( ( ph /\ dom OrdIso ( R , X ) C_ dom OrdIso ( S , Y ) ) -> A e. _V )
10		fpwwe2.3	. . . . . 6 |- ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A )
11	10	adantlr 751	. . . . 5 |- ( ( ( ph /\ dom OrdIso ( R , X ) C_ dom OrdIso ( S , Y ) ) /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A )
12		fpwwe2lem10.4	. . . . . 6 |- ( ph -> X W R )
13	12	adantr 481	. . . . 5 |- ( ( ph /\ dom OrdIso ( R , X ) C_ dom OrdIso ( S , Y ) ) -> X W R )
14		fpwwe2lem10.6	. . . . . 6 |- ( ph -> Y W S )
15	14	adantr 481	. . . . 5 |- ( ( ph /\ dom OrdIso ( R , X ) C_ dom OrdIso ( S , Y ) ) -> Y W S )
16		simpr 477	. . . . 5 |- ( ( ph /\ dom OrdIso ( R , X ) C_ dom OrdIso ( S , Y ) ) -> dom OrdIso ( R , X ) C_ dom OrdIso ( S , Y ) )
17	7, 9, 11, 13, 15, 1, 3, 16	fpwwe2lem9 9460	. . . 4 |- ( ( ph /\ dom OrdIso ( R , X ) C_ dom OrdIso ( S , Y ) ) -> ( X C_ Y /\ R = ( S i^i ( Y X. X ) ) ) )
18	17	ex 450	. . 3 |- ( ph -> ( dom OrdIso ( R , X ) C_ dom OrdIso ( S , Y ) -> ( X C_ Y /\ R = ( S i^i ( Y X. X ) ) ) ) )
19	8	adantr 481	. . . . 5 |- ( ( ph /\ dom OrdIso ( S , Y ) C_ dom OrdIso ( R , X ) ) -> A e. _V )
20	10	adantlr 751	. . . . 5 |- ( ( ( ph /\ dom OrdIso ( S , Y ) C_ dom OrdIso ( R , X ) ) /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A )
21	14	adantr 481	. . . . 5 |- ( ( ph /\ dom OrdIso ( S , Y ) C_ dom OrdIso ( R , X ) ) -> Y W S )
22	12	adantr 481	. . . . 5 |- ( ( ph /\ dom OrdIso ( S , Y ) C_ dom OrdIso ( R , X ) ) -> X W R )
23		simpr 477	. . . . 5 |- ( ( ph /\ dom OrdIso ( S , Y ) C_ dom OrdIso ( R , X ) ) -> dom OrdIso ( S , Y ) C_ dom OrdIso ( R , X ) )
24	7, 19, 20, 21, 22, 3, 1, 23	fpwwe2lem9 9460	. . . 4 |- ( ( ph /\ dom OrdIso ( S , Y ) C_ dom OrdIso ( R , X ) ) -> ( Y C_ X /\ S = ( R i^i ( X X. Y ) ) ) )
25	24	ex 450	. . 3 |- ( ph -> ( dom OrdIso ( S , Y ) C_ dom OrdIso ( R , X ) -> ( Y C_ X /\ S = ( R i^i ( X X. Y ) ) ) ) )
26	18, 25	orim12d 883	. 2 |- ( ph -> ( ( dom OrdIso ( R , X ) C_ dom OrdIso ( S , Y ) \/ dom OrdIso ( S , Y ) C_ dom OrdIso ( R , X ) ) -> ( ( X C_ Y /\ R = ( S i^i ( Y X. X ) ) ) \/ ( Y C_ X /\ S = ( R i^i ( X X. Y ) ) ) ) ) )
27	6, 26	mpi 20	1 |- ( ph -> ( ( X C_ Y /\ R = ( S i^i ( Y X. X ) ) ) \/ ( Y C_ X /\ S = ( R i^i ( X X. Y ) ) ) ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [.wsbc 3435   i^i cin 3573   C_ wss 3574   {csn 4177   class class class wbr 4653   {copab 4712   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   Ord word 5722  (class class class)co 6650  OrdIsocoi 8414

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-wrecs 7407  df-recs 7468  df-oi 8415

This theorem is referenced by:  fpwwe2lem11  9462  fpwwe2lem12  9463  fpwwe2  9465