Theorem fpwwe2lem7 9458

Description: Lemma for fpwwe2 9465. (Contributed by Mario Carneiro, 18-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 )
fpwwe2lem9.x	|- ( ph -> X W R )
fpwwe2lem9.y	|- ( ph -> Y W S )
fpwwe2lem9.m	|- M = OrdIso ( R , X )
fpwwe2lem9.n	|- N = OrdIso ( S , Y )
fpwwe2lem7.1	|- ( ph -> B e. dom M )
fpwwe2lem7.2	|- ( ph -> B e. dom N )
fpwwe2lem7.3	|- ( ph -> ( M |` B ) = ( N |` B ) )

Assertion

Ref	Expression
fpwwe2lem7	|- ( ( ph /\ C R ( M ` B ) ) -> ( C S ( N ` B ) /\ ( D R ( M ` B ) -> ( C R D <-> C S D ) ) ) )

Distinct variable groups:   y, u, B   u, r, x, y, F   X, r, u, x, y   M, r, u, x, y   N, 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)   B( x, r)   C( x, y, u, r)   D( x, y, u, r)

Proof of Theorem fpwwe2lem7

Step	Hyp	Ref	Expression
1		fpwwe2lem9.y	. . . . . . . 8 |- ( ph -> Y W S )
2		fpwwe2.1	. . . . . . . . . 10 |- 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 ) ) }
3	2	relopabi 5245	. . . . . . . . 9 |- Rel W
4	3	brrelexi 5158	. . . . . . . 8 |- ( Y W S -> Y e. _V )
5	1, 4	syl 17	. . . . . . 7 |- ( ph -> Y e. _V )
6		fpwwe2.2	. . . . . . . . . . 11 |- ( ph -> A e. _V )
7	2, 6	fpwwe2lem2 9454	. . . . . . . . . 10 |- ( ph -> ( Y W S <-> ( ( Y C_ A /\ S C_ ( Y X. Y ) ) /\ ( S We Y /\ A. y e. Y [. ( `' S " { y } ) / u ]. ( u F ( S i^i ( u X. u ) ) ) = y ) ) ) )
8	1, 7	mpbid 222	. . . . . . . . 9 |- ( ph -> ( ( Y C_ A /\ S C_ ( Y X. Y ) ) /\ ( S We Y /\ A. y e. Y [. ( `' S " { y } ) / u ]. ( u F ( S i^i ( u X. u ) ) ) = y ) ) )
9	8	simprd 479	. . . . . . . 8 |- ( ph -> ( S We Y /\ A. y e. Y [. ( `' S " { y } ) / u ]. ( u F ( S i^i ( u X. u ) ) ) = y ) )
10	9	simpld 475	. . . . . . 7 |- ( ph -> S We Y )
11		fpwwe2lem9.n	. . . . . . . 8 |- N = OrdIso ( S , Y )
12	11	oiiso 8442	. . . . . . 7 |- ( ( Y e. _V /\ S We Y ) -> N Isom _E , S ( dom N , Y ) )
13	5, 10, 12	syl2anc 693	. . . . . 6 |- ( ph -> N Isom _E , S ( dom N , Y ) )
14	13	adantr 481	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> N Isom _E , S ( dom N , Y ) )
15		isof1o 6573	. . . . 5 |- ( N Isom _E , S ( dom N , Y ) -> N : dom N -1-1-onto-> Y )
16	14, 15	syl 17	. . . 4 |- ( ( ph /\ C R ( M ` B ) ) -> N : dom N -1-1-onto-> Y )
17		fpwwe2.3	. . . . . 6 |- ( ( ph /\ ( x C_ A /\ r C_ ( x X. x ) /\ r We x ) ) -> ( x F r ) e. A )
18		fpwwe2lem9.x	. . . . . 6 |- ( ph -> X W R )
19		fpwwe2lem9.m	. . . . . 6 |- M = OrdIso ( R , X )
20		fpwwe2lem7.1	. . . . . 6 |- ( ph -> B e. dom M )
21		fpwwe2lem7.2	. . . . . 6 |- ( ph -> B e. dom N )
22		fpwwe2lem7.3	. . . . . 6 |- ( ph -> ( M |` B ) = ( N |` B ) )
23	2, 6, 17, 18, 1, 19, 11, 20, 21, 22	fpwwe2lem6 9457	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> ( C e. X /\ C e. Y /\ ( `' M ` C ) = ( `' N ` C ) ) )
24	23	simp2d 1074	. . . 4 |- ( ( ph /\ C R ( M ` B ) ) -> C e. Y )
25		f1ocnvfv2 6533	. . . 4 |- ( ( N : dom N -1-1-onto-> Y /\ C e. Y ) -> ( N ` ( `' N ` C ) ) = C )
26	16, 24, 25	syl2anc 693	. . 3 |- ( ( ph /\ C R ( M ` B ) ) -> ( N ` ( `' N ` C ) ) = C )
27	23	simp3d 1075	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) = ( `' N ` C ) )
28	3	brrelexi 5158	. . . . . . . . . . . 12 |- ( X W R -> X e. _V )
29	18, 28	syl 17	. . . . . . . . . . 11 |- ( ph -> X e. _V )
30	2, 6	fpwwe2lem2 9454	. . . . . . . . . . . . . 14 |- ( ph -> ( X W 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 ) ) ) )
31	18, 30	mpbid 222	. . . . . . . . . . . . 13 |- ( ph -> ( ( 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 ) ) )
32	31	simprd 479	. . . . . . . . . . . 12 |- ( ph -> ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) )
33	32	simpld 475	. . . . . . . . . . 11 |- ( ph -> R We X )
34	19	oiiso 8442	. . . . . . . . . . 11 |- ( ( X e. _V /\ R We X ) -> M Isom _E , R ( dom M , X ) )
35	29, 33, 34	syl2anc 693	. . . . . . . . . 10 |- ( ph -> M Isom _E , R ( dom M , X ) )
36	35	adantr 481	. . . . . . . . 9 |- ( ( ph /\ C R ( M ` B ) ) -> M Isom _E , R ( dom M , X ) )
37		isof1o 6573	. . . . . . . . 9 |- ( M Isom _E , R ( dom M , X ) -> M : dom M -1-1-onto-> X )
38	36, 37	syl 17	. . . . . . . 8 |- ( ( ph /\ C R ( M ` B ) ) -> M : dom M -1-1-onto-> X )
39	23	simp1d 1073	. . . . . . . 8 |- ( ( ph /\ C R ( M ` B ) ) -> C e. X )
40		f1ocnvfv2 6533	. . . . . . . 8 |- ( ( M : dom M -1-1-onto-> X /\ C e. X ) -> ( M ` ( `' M ` C ) ) = C )
41	38, 39, 40	syl2anc 693	. . . . . . 7 |- ( ( ph /\ C R ( M ` B ) ) -> ( M ` ( `' M ` C ) ) = C )
42		simpr 477	. . . . . . 7 |- ( ( ph /\ C R ( M ` B ) ) -> C R ( M ` B ) )
43	41, 42	eqbrtrd 4675	. . . . . 6 |- ( ( ph /\ C R ( M ` B ) ) -> ( M ` ( `' M ` C ) ) R ( M ` B ) )
44		f1ocnv 6149	. . . . . . . . 9 |- ( M : dom M -1-1-onto-> X -> `' M : X -1-1-onto-> dom M )
45		f1of 6137	. . . . . . . . 9 |- ( `' M : X -1-1-onto-> dom M -> `' M : X --> dom M )
46	38, 44, 45	3syl 18	. . . . . . . 8 |- ( ( ph /\ C R ( M ` B ) ) -> `' M : X --> dom M )
47	46, 39	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) e. dom M )
48	20	adantr 481	. . . . . . 7 |- ( ( ph /\ C R ( M ` B ) ) -> B e. dom M )
49		isorel 6576	. . . . . . 7 |- ( ( M Isom _E , R ( dom M , X ) /\ ( ( `' M ` C ) e. dom M /\ B e. dom M ) ) -> ( ( `' M ` C ) _E B <-> ( M ` ( `' M ` C ) ) R ( M ` B ) ) )
50	36, 47, 48, 49	syl12anc 1324	. . . . . 6 |- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' M ` C ) _E B <-> ( M ` ( `' M ` C ) ) R ( M ` B ) ) )
51	43, 50	mpbird 247	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) _E B )
52	27, 51	eqbrtrrd 4677	. . . 4 |- ( ( ph /\ C R ( M ` B ) ) -> ( `' N ` C ) _E B )
53		f1ocnv 6149	. . . . . . 7 |- ( N : dom N -1-1-onto-> Y -> `' N : Y -1-1-onto-> dom N )
54		f1of 6137	. . . . . . 7 |- ( `' N : Y -1-1-onto-> dom N -> `' N : Y --> dom N )
55	16, 53, 54	3syl 18	. . . . . 6 |- ( ( ph /\ C R ( M ` B ) ) -> `' N : Y --> dom N )
56	55, 24	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> ( `' N ` C ) e. dom N )
57	21	adantr 481	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> B e. dom N )
58		isorel 6576	. . . . 5 |- ( ( N Isom _E , S ( dom N , Y ) /\ ( ( `' N ` C ) e. dom N /\ B e. dom N ) ) -> ( ( `' N ` C ) _E B <-> ( N ` ( `' N ` C ) ) S ( N ` B ) ) )
59	14, 56, 57, 58	syl12anc 1324	. . . 4 |- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' N ` C ) _E B <-> ( N ` ( `' N ` C ) ) S ( N ` B ) ) )
60	52, 59	mpbid 222	. . 3 |- ( ( ph /\ C R ( M ` B ) ) -> ( N ` ( `' N ` C ) ) S ( N ` B ) )
61	26, 60	eqbrtrrd 4677	. 2 |- ( ( ph /\ C R ( M ` B ) ) -> C S ( N ` B ) )
62	27	adantrr 753	. . . . 5 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( `' M ` C ) = ( `' N ` C ) )
63	2, 6, 17, 18, 1, 19, 11, 20, 21, 22	fpwwe2lem6 9457	. . . . . . 7 |- ( ( ph /\ D R ( M ` B ) ) -> ( D e. X /\ D e. Y /\ ( `' M ` D ) = ( `' N ` D ) ) )
64	63	simp3d 1075	. . . . . 6 |- ( ( ph /\ D R ( M ` B ) ) -> ( `' M ` D ) = ( `' N ` D ) )
65	64	adantrl 752	. . . . 5 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( `' M ` D ) = ( `' N ` D ) )
66	62, 65	breq12d 4666	. . . 4 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( ( `' M ` C ) _E ( `' M ` D ) <-> ( `' N ` C ) _E ( `' N ` D ) ) )
67	35	adantr 481	. . . . . 6 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> M Isom _E , R ( dom M , X ) )
68		isocnv 6580	. . . . . 6 |- ( M Isom _E , R ( dom M , X ) -> `' M Isom R , _E ( X , dom M ) )
69	67, 68	syl 17	. . . . 5 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> `' M Isom R , _E ( X , dom M ) )
70	39	adantrr 753	. . . . 5 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> C e. X )
71	31	simpld 475	. . . . . . . . . 10 |- ( ph -> ( X C_ A /\ R C_ ( X X. X ) ) )
72	71	simprd 479	. . . . . . . . 9 |- ( ph -> R C_ ( X X. X ) )
73	72	ssbrd 4696	. . . . . . . 8 |- ( ph -> ( D R ( M ` B ) -> D ( X X. X ) ( M ` B ) ) )
74	73	imp 445	. . . . . . 7 |- ( ( ph /\ D R ( M ` B ) ) -> D ( X X. X ) ( M ` B ) )
75		brxp 5147	. . . . . . . 8 |- ( D ( X X. X ) ( M ` B ) <-> ( D e. X /\ ( M ` B ) e. X ) )
76	75	simplbi 476	. . . . . . 7 |- ( D ( X X. X ) ( M ` B ) -> D e. X )
77	74, 76	syl 17	. . . . . 6 |- ( ( ph /\ D R ( M ` B ) ) -> D e. X )
78	77	adantrl 752	. . . . 5 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> D e. X )
79		isorel 6576	. . . . 5 |- ( ( `' M Isom R , _E ( X , dom M ) /\ ( C e. X /\ D e. X ) ) -> ( C R D <-> ( `' M ` C ) _E ( `' M ` D ) ) )
80	69, 70, 78, 79	syl12anc 1324	. . . 4 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( C R D <-> ( `' M ` C ) _E ( `' M ` D ) ) )
81	13	adantr 481	. . . . . 6 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> N Isom _E , S ( dom N , Y ) )
82		isocnv 6580	. . . . . 6 |- ( N Isom _E , S ( dom N , Y ) -> `' N Isom S , _E ( Y , dom N ) )
83	81, 82	syl 17	. . . . 5 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> `' N Isom S , _E ( Y , dom N ) )
84	24	adantrr 753	. . . . 5 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> C e. Y )
85	63	simp2d 1074	. . . . . 6 |- ( ( ph /\ D R ( M ` B ) ) -> D e. Y )
86	85	adantrl 752	. . . . 5 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> D e. Y )
87		isorel 6576	. . . . 5 |- ( ( `' N Isom S , _E ( Y , dom N ) /\ ( C e. Y /\ D e. Y ) ) -> ( C S D <-> ( `' N ` C ) _E ( `' N ` D ) ) )
88	83, 84, 86, 87	syl12anc 1324	. . . 4 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( C S D <-> ( `' N ` C ) _E ( `' N ` D ) ) )
89	66, 80, 88	3bitr4d 300	. . 3 |- ( ( ph /\ ( C R ( M ` B ) /\ D R ( M ` B ) ) ) -> ( C R D <-> C S D ) )
90	89	expr 643	. 2 |- ( ( ph /\ C R ( M ` B ) ) -> ( D R ( M ` B ) -> ( C R D <-> C S D ) ) )
91	61, 90	jca 554	1 |- ( ( ph /\ C R ( M ` B ) ) -> ( C S ( N ` B ) /\ ( D R ( M ` B ) -> ( C R D <-> C S D ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ 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   _E cep 5028   We wwe 5072   X. cxp 5112   `'ccnv 5113   dom cdm 5114   |` cres 5116   "cima 5117   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (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:  fpwwe2lem8  9459