Theorem fpwwe2lem6 9457

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
fpwwe2lem6	|- ( ( ph /\ C R ( M ` B ) ) -> ( C e. X /\ C e. Y /\ ( `' M ` C ) = ( `' N ` C ) ) )

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)

Proof of Theorem fpwwe2lem6

Step	Hyp	Ref	Expression
1		fpwwe2lem9.x	. . . . . . . 8 |- ( ph -> X W R )
2		fpwwe2.1	. . . . . . . . 9 |- 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		fpwwe2.2	. . . . . . . . 9 |- ( ph -> A e. _V )
4	2, 3	fpwwe2lem2 9454	. . . . . . . 8 |- ( 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 ) ) ) )
5	1, 4	mpbid 222	. . . . . . 7 |- ( 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 ) ) )
6	5	simpld 475	. . . . . 6 |- ( ph -> ( X C_ A /\ R C_ ( X X. X ) ) )
7	6	simprd 479	. . . . 5 |- ( ph -> R C_ ( X X. X ) )
8	7	ssbrd 4696	. . . 4 |- ( ph -> ( C R ( M ` B ) -> C ( X X. X ) ( M ` B ) ) )
9		brxp 5147	. . . . 5 |- ( C ( X X. X ) ( M ` B ) <-> ( C e. X /\ ( M ` B ) e. X ) )
10	9	simplbi 476	. . . 4 |- ( C ( X X. X ) ( M ` B ) -> C e. X )
11	8, 10	syl6 35	. . 3 |- ( ph -> ( C R ( M ` B ) -> C e. X ) )
12	11	imp 445	. 2 |- ( ( ph /\ C R ( M ` B ) ) -> C e. X )
13		imassrn 5477	. . . 4 |- ( N " B ) C_ ran N
14		fpwwe2lem9.y	. . . . . . . . 9 |- ( ph -> Y W S )
15	2	relopabi 5245	. . . . . . . . . 10 |- Rel W
16	15	brrelexi 5158	. . . . . . . . 9 |- ( Y W S -> Y e. _V )
17	14, 16	syl 17	. . . . . . . 8 |- ( ph -> Y e. _V )
18	2, 3	fpwwe2lem2 9454	. . . . . . . . . . 11 |- ( 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 ) ) ) )
19	14, 18	mpbid 222	. . . . . . . . . 10 |- ( 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 ) ) )
20	19	simprd 479	. . . . . . . . 9 |- ( ph -> ( S We Y /\ A. y e. Y [. ( `' S " { y } ) / u ]. ( u F ( S i^i ( u X. u ) ) ) = y ) )
21	20	simpld 475	. . . . . . . 8 |- ( ph -> S We Y )
22		fpwwe2lem9.n	. . . . . . . . 9 |- N = OrdIso ( S , Y )
23	22	oiiso 8442	. . . . . . . 8 |- ( ( Y e. _V /\ S We Y ) -> N Isom _E , S ( dom N , Y ) )
24	17, 21, 23	syl2anc 693	. . . . . . 7 |- ( ph -> N Isom _E , S ( dom N , Y ) )
25	24	adantr 481	. . . . . 6 |- ( ( ph /\ C R ( M ` B ) ) -> N Isom _E , S ( dom N , Y ) )
26		isof1o 6573	. . . . . 6 |- ( N Isom _E , S ( dom N , Y ) -> N : dom N -1-1-onto-> Y )
27	25, 26	syl 17	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> N : dom N -1-1-onto-> Y )
28		f1ofo 6144	. . . . 5 |- ( N : dom N -1-1-onto-> Y -> N : dom N -onto-> Y )
29		forn 6118	. . . . 5 |- ( N : dom N -onto-> Y -> ran N = Y )
30	27, 28, 29	3syl 18	. . . 4 |- ( ( ph /\ C R ( M ` B ) ) -> ran N = Y )
31	13, 30	syl5sseq 3653	. . 3 |- ( ( ph /\ C R ( M ` B ) ) -> ( N " B ) C_ Y )
32	15	brrelexi 5158	. . . . . . . . . . . . . 14 |- ( X W R -> X e. _V )
33	1, 32	syl 17	. . . . . . . . . . . . 13 |- ( ph -> X e. _V )
34	5	simprd 479	. . . . . . . . . . . . . 14 |- ( ph -> ( R We X /\ A. y e. X [. ( `' R " { y } ) / u ]. ( u F ( R i^i ( u X. u ) ) ) = y ) )
35	34	simpld 475	. . . . . . . . . . . . 13 |- ( ph -> R We X )
36		fpwwe2lem9.m	. . . . . . . . . . . . . 14 |- M = OrdIso ( R , X )
37	36	oiiso 8442	. . . . . . . . . . . . 13 |- ( ( X e. _V /\ R We X ) -> M Isom _E , R ( dom M , X ) )
38	33, 35, 37	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> M Isom _E , R ( dom M , X ) )
39	38	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ C R ( M ` B ) ) -> M Isom _E , R ( dom M , X ) )
40		isof1o 6573	. . . . . . . . . . 11 |- ( M Isom _E , R ( dom M , X ) -> M : dom M -1-1-onto-> X )
41	39, 40	syl 17	. . . . . . . . . 10 |- ( ( ph /\ C R ( M ` B ) ) -> M : dom M -1-1-onto-> X )
42		f1ocnvfv2 6533	. . . . . . . . . 10 |- ( ( M : dom M -1-1-onto-> X /\ C e. X ) -> ( M ` ( `' M ` C ) ) = C )
43	41, 12, 42	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ C R ( M ` B ) ) -> ( M ` ( `' M ` C ) ) = C )
44		simpr 477	. . . . . . . . 9 |- ( ( ph /\ C R ( M ` B ) ) -> C R ( M ` B ) )
45	43, 44	eqbrtrd 4675	. . . . . . . 8 |- ( ( ph /\ C R ( M ` B ) ) -> ( M ` ( `' M ` C ) ) R ( M ` B ) )
46		f1ocnv 6149	. . . . . . . . . . 11 |- ( M : dom M -1-1-onto-> X -> `' M : X -1-1-onto-> dom M )
47		f1of 6137	. . . . . . . . . . 11 |- ( `' M : X -1-1-onto-> dom M -> `' M : X --> dom M )
48	41, 46, 47	3syl 18	. . . . . . . . . 10 |- ( ( ph /\ C R ( M ` B ) ) -> `' M : X --> dom M )
49	48, 12	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) e. dom M )
50		fpwwe2lem7.1	. . . . . . . . . 10 |- ( ph -> B e. dom M )
51	50	adantr 481	. . . . . . . . 9 |- ( ( ph /\ C R ( M ` B ) ) -> B e. dom M )
52		isorel 6576	. . . . . . . . 9 |- ( ( 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 ) ) )
53	39, 49, 51, 52	syl12anc 1324	. . . . . . . 8 |- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' M ` C ) _E B <-> ( M ` ( `' M ` C ) ) R ( M ` B ) ) )
54	45, 53	mpbird 247	. . . . . . 7 |- ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) _E B )
55		epelg 5030	. . . . . . . 8 |- ( B e. dom M -> ( ( `' M ` C ) _E B <-> ( `' M ` C ) e. B ) )
56	51, 55	syl 17	. . . . . . 7 |- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' M ` C ) _E B <-> ( `' M ` C ) e. B ) )
57	54, 56	mpbid 222	. . . . . 6 |- ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) e. B )
58		ffn 6045	. . . . . . 7 |- ( `' M : X --> dom M -> `' M Fn X )
59		elpreima 6337	. . . . . . 7 |- ( `' M Fn X -> ( C e. ( `' `' M " B ) <-> ( C e. X /\ ( `' M ` C ) e. B ) ) )
60	48, 58, 59	3syl 18	. . . . . 6 |- ( ( ph /\ C R ( M ` B ) ) -> ( C e. ( `' `' M " B ) <-> ( C e. X /\ ( `' M ` C ) e. B ) ) )
61	12, 57, 60	mpbir2and 957	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> C e. ( `' `' M " B ) )
62		imacnvcnv 5599	. . . . 5 |- ( `' `' M " B ) = ( M " B )
63	61, 62	syl6eleq 2711	. . . 4 |- ( ( ph /\ C R ( M ` B ) ) -> C e. ( M " B ) )
64		fpwwe2lem7.3	. . . . . . 7 |- ( ph -> ( M |` B ) = ( N |` B ) )
65	64	adantr 481	. . . . . 6 |- ( ( ph /\ C R ( M ` B ) ) -> ( M |` B ) = ( N |` B ) )
66	65	rneqd 5353	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> ran ( M |` B ) = ran ( N |` B ) )
67		df-ima 5127	. . . . 5 |- ( M " B ) = ran ( M |` B )
68		df-ima 5127	. . . . 5 |- ( N " B ) = ran ( N |` B )
69	66, 67, 68	3eqtr4g 2681	. . . 4 |- ( ( ph /\ C R ( M ` B ) ) -> ( M " B ) = ( N " B ) )
70	63, 69	eleqtrd 2703	. . 3 |- ( ( ph /\ C R ( M ` B ) ) -> C e. ( N " B ) )
71	31, 70	sseldd 3604	. 2 |- ( ( ph /\ C R ( M ` B ) ) -> C e. Y )
72	65	cnveqd 5298	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> `' ( M |` B ) = `' ( N |` B ) )
73		dff1o3 6143	. . . . . . 7 |- ( M : dom M -1-1-onto-> X <-> ( M : dom M -onto-> X /\ Fun `' M ) )
74	73	simprbi 480	. . . . . 6 |- ( M : dom M -1-1-onto-> X -> Fun `' M )
75		funcnvres 5967	. . . . . 6 |- ( Fun `' M -> `' ( M |` B ) = ( `' M |` ( M " B ) ) )
76	41, 74, 75	3syl 18	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> `' ( M |` B ) = ( `' M |` ( M " B ) ) )
77		dff1o3 6143	. . . . . . 7 |- ( N : dom N -1-1-onto-> Y <-> ( N : dom N -onto-> Y /\ Fun `' N ) )
78	77	simprbi 480	. . . . . 6 |- ( N : dom N -1-1-onto-> Y -> Fun `' N )
79		funcnvres 5967	. . . . . 6 |- ( Fun `' N -> `' ( N |` B ) = ( `' N |` ( N " B ) ) )
80	27, 78, 79	3syl 18	. . . . 5 |- ( ( ph /\ C R ( M ` B ) ) -> `' ( N |` B ) = ( `' N |` ( N " B ) ) )
81	72, 76, 80	3eqtr3d 2664	. . . 4 |- ( ( ph /\ C R ( M ` B ) ) -> ( `' M |` ( M " B ) ) = ( `' N |` ( N " B ) ) )
82	81	fveq1d 6193	. . 3 |- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' M |` ( M " B ) ) ` C ) = ( ( `' N |` ( N " B ) ) ` C ) )
83		fvres 6207	. . . 4 |- ( C e. ( M " B ) -> ( ( `' M |` ( M " B ) ) ` C ) = ( `' M ` C ) )
84	63, 83	syl 17	. . 3 |- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' M |` ( M " B ) ) ` C ) = ( `' M ` C ) )
85		fvres 6207	. . . 4 |- ( C e. ( N " B ) -> ( ( `' N |` ( N " B ) ) ` C ) = ( `' N ` C ) )
86	70, 85	syl 17	. . 3 |- ( ( ph /\ C R ( M ` B ) ) -> ( ( `' N |` ( N " B ) ) ` C ) = ( `' N ` C ) )
87	82, 84, 86	3eqtr3d 2664	. 2 |- ( ( ph /\ C R ( M ` B ) ) -> ( `' M ` C ) = ( `' N ` C ) )
88	12, 71, 87	3jca 1242	1 |- ( ( ph /\ C R ( M ` B ) ) -> ( C e. X /\ C e. Y /\ ( `' M ` C ) = ( `' N ` C ) ) )

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   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   -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:  fpwwe2lem7  9458