Theorem isofrlem 6590

Description: Lemma for isofr 6592. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)

Hypotheses

Ref	Expression
isofrlem.1	|- ( ph -> H Isom R , S ( A , B ) )
isofrlem.2	|- ( ph -> ( H " x ) e. _V )

Assertion

Ref	Expression
isofrlem	|- ( ph -> ( S Fr B -> R Fr A ) )

Distinct variable groups:   x, A   x, B   x, H   ph, x   x, R   x, S

Proof of Theorem isofrlem

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isofrlem.1	. . . . . . 7 |- ( ph -> H Isom R , S ( A , B ) )
2		isof1o 6573	. . . . . . 7 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
3	1, 2	syl 17	. . . . . 6 |- ( ph -> H : A -1-1-onto-> B )
4		f1ofn 6138	. . . . . . . 8 |- ( H : A -1-1-onto-> B -> H Fn A )
5		n0 3931	. . . . . . . . . 10 |- ( x =/= (/) <-> E. y y e. x )
6		fnfvima 6496	. . . . . . . . . . . . 13 |- ( ( H Fn A /\ x C_ A /\ y e. x ) -> ( H ` y ) e. ( H " x ) )
7		ne0i 3921	. . . . . . . . . . . . 13 |- ( ( H ` y ) e. ( H " x ) -> ( H " x ) =/= (/) )
8	6, 7	syl 17	. . . . . . . . . . . 12 |- ( ( H Fn A /\ x C_ A /\ y e. x ) -> ( H " x ) =/= (/) )
9	8	3expia 1267	. . . . . . . . . . 11 |- ( ( H Fn A /\ x C_ A ) -> ( y e. x -> ( H " x ) =/= (/) ) )
10	9	exlimdv 1861	. . . . . . . . . 10 |- ( ( H Fn A /\ x C_ A ) -> ( E. y y e. x -> ( H " x ) =/= (/) ) )
11	5, 10	syl5bi 232	. . . . . . . . 9 |- ( ( H Fn A /\ x C_ A ) -> ( x =/= (/) -> ( H " x ) =/= (/) ) )
12	11	expimpd 629	. . . . . . . 8 |- ( H Fn A -> ( ( x C_ A /\ x =/= (/) ) -> ( H " x ) =/= (/) ) )
13	4, 12	syl 17	. . . . . . 7 |- ( H : A -1-1-onto-> B -> ( ( x C_ A /\ x =/= (/) ) -> ( H " x ) =/= (/) ) )
14		f1ofo 6144	. . . . . . . 8 |- ( H : A -1-1-onto-> B -> H : A -onto-> B )
15		imassrn 5477	. . . . . . . . 9 |- ( H " x ) C_ ran H
16		forn 6118	. . . . . . . . 9 |- ( H : A -onto-> B -> ran H = B )
17	15, 16	syl5sseq 3653	. . . . . . . 8 |- ( H : A -onto-> B -> ( H " x ) C_ B )
18	14, 17	syl 17	. . . . . . 7 |- ( H : A -1-1-onto-> B -> ( H " x ) C_ B )
19	13, 18	jctild 566	. . . . . 6 |- ( H : A -1-1-onto-> B -> ( ( x C_ A /\ x =/= (/) ) -> ( ( H " x ) C_ B /\ ( H " x ) =/= (/) ) ) )
20	3, 19	syl 17	. . . . 5 |- ( ph -> ( ( x C_ A /\ x =/= (/) ) -> ( ( H " x ) C_ B /\ ( H " x ) =/= (/) ) ) )
21		dffr3 5498	. . . . . 6 |- ( S Fr B <-> A. z ( ( z C_ B /\ z =/= (/) ) -> E. w e. z ( z i^i ( `' S " { w } ) ) = (/) ) )
22		isofrlem.2	. . . . . . 7 |- ( ph -> ( H " x ) e. _V )
23		sseq1 3626	. . . . . . . . . 10 |- ( z = ( H " x ) -> ( z C_ B <-> ( H " x ) C_ B ) )
24		neeq1 2856	. . . . . . . . . 10 |- ( z = ( H " x ) -> ( z =/= (/) <-> ( H " x ) =/= (/) ) )
25	23, 24	anbi12d 747	. . . . . . . . 9 |- ( z = ( H " x ) -> ( ( z C_ B /\ z =/= (/) ) <-> ( ( H " x ) C_ B /\ ( H " x ) =/= (/) ) ) )
26		ineq1 3807	. . . . . . . . . . 11 |- ( z = ( H " x ) -> ( z i^i ( `' S " { w } ) ) = ( ( H " x ) i^i ( `' S " { w } ) ) )
27	26	eqeq1d 2624	. . . . . . . . . 10 |- ( z = ( H " x ) -> ( ( z i^i ( `' S " { w } ) ) = (/) <-> ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) )
28	27	rexeqbi1dv 3147	. . . . . . . . 9 |- ( z = ( H " x ) -> ( E. w e. z ( z i^i ( `' S " { w } ) ) = (/) <-> E. w e. ( H " x ) ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) )
29	25, 28	imbi12d 334	. . . . . . . 8 |- ( z = ( H " x ) -> ( ( ( z C_ B /\ z =/= (/) ) -> E. w e. z ( z i^i ( `' S " { w } ) ) = (/) ) <-> ( ( ( H " x ) C_ B /\ ( H " x ) =/= (/) ) -> E. w e. ( H " x ) ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) ) )
30	29	spcgv 3293	. . . . . . 7 |- ( ( H " x ) e. _V -> ( A. z ( ( z C_ B /\ z =/= (/) ) -> E. w e. z ( z i^i ( `' S " { w } ) ) = (/) ) -> ( ( ( H " x ) C_ B /\ ( H " x ) =/= (/) ) -> E. w e. ( H " x ) ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) ) )
31	22, 30	syl 17	. . . . . 6 |- ( ph -> ( A. z ( ( z C_ B /\ z =/= (/) ) -> E. w e. z ( z i^i ( `' S " { w } ) ) = (/) ) -> ( ( ( H " x ) C_ B /\ ( H " x ) =/= (/) ) -> E. w e. ( H " x ) ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) ) )
32	21, 31	syl5bi 232	. . . . 5 |- ( ph -> ( S Fr B -> ( ( ( H " x ) C_ B /\ ( H " x ) =/= (/) ) -> E. w e. ( H " x ) ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) ) )
33	20, 32	syl5d 73	. . . 4 |- ( ph -> ( S Fr B -> ( ( x C_ A /\ x =/= (/) ) -> E. w e. ( H " x ) ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) ) )
34	3	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ x C_ A ) -> H : A -1-1-onto-> B )
35		f1ofun 6139	. . . . . . . . . . 11 |- ( H : A -1-1-onto-> B -> Fun H )
36	34, 35	syl 17	. . . . . . . . . 10 |- ( ( ph /\ x C_ A ) -> Fun H )
37		simpl 473	. . . . . . . . . 10 |- ( ( w e. ( H " x ) /\ ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) -> w e. ( H " x ) )
38		fvelima 6248	. . . . . . . . . 10 |- ( ( Fun H /\ w e. ( H " x ) ) -> E. y e. x ( H ` y ) = w )
39	36, 37, 38	syl2an 494	. . . . . . . . 9 |- ( ( ( ph /\ x C_ A ) /\ ( w e. ( H " x ) /\ ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) ) -> E. y e. x ( H ` y ) = w )
40		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( w e. ( H " x ) /\ ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) -> ( ( H " x ) i^i ( `' S " { w } ) ) = (/) )
41		ssel 3597	. . . . . . . . . . . . . . . . . . 19 |- ( x C_ A -> ( y e. x -> y e. A ) )
42	41	imdistani 726	. . . . . . . . . . . . . . . . . 18 |- ( ( x C_ A /\ y e. x ) -> ( x C_ A /\ y e. A ) )
43		isomin 6587	. . . . . . . . . . . . . . . . . 18 |- ( ( H Isom R , S ( A , B ) /\ ( x C_ A /\ y e. A ) ) -> ( ( x i^i ( `' R " { y } ) ) = (/) <-> ( ( H " x ) i^i ( `' S " { ( H ` y ) } ) ) = (/) ) )
44	1, 42, 43	syl2an 494	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( x C_ A /\ y e. x ) ) -> ( ( x i^i ( `' R " { y } ) ) = (/) <-> ( ( H " x ) i^i ( `' S " { ( H ` y ) } ) ) = (/) ) )
45		sneq 4187	. . . . . . . . . . . . . . . . . . . 20 |- ( ( H ` y ) = w -> { ( H ` y ) } = { w } )
46	45	imaeq2d 5466	. . . . . . . . . . . . . . . . . . 19 |- ( ( H ` y ) = w -> ( `' S " { ( H ` y ) } ) = ( `' S " { w } ) )
47	46	ineq2d 3814	. . . . . . . . . . . . . . . . . 18 |- ( ( H ` y ) = w -> ( ( H " x ) i^i ( `' S " { ( H ` y ) } ) ) = ( ( H " x ) i^i ( `' S " { w } ) ) )
48	47	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 |- ( ( H ` y ) = w -> ( ( ( H " x ) i^i ( `' S " { ( H ` y ) } ) ) = (/) <-> ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) )
49	44, 48	sylan9bb 736	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( x C_ A /\ y e. x ) ) /\ ( H ` y ) = w ) -> ( ( x i^i ( `' R " { y } ) ) = (/) <-> ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) )
50	40, 49	syl5ibr 236	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( x C_ A /\ y e. x ) ) /\ ( H ` y ) = w ) -> ( ( w e. ( H " x ) /\ ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) -> ( x i^i ( `' R " { y } ) ) = (/) ) )
51	50	exp42 639	. . . . . . . . . . . . . 14 |- ( ph -> ( x C_ A -> ( y e. x -> ( ( H ` y ) = w -> ( ( w e. ( H " x ) /\ ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) -> ( x i^i ( `' R " { y } ) ) = (/) ) ) ) ) )
52	51	imp 445	. . . . . . . . . . . . 13 |- ( ( ph /\ x C_ A ) -> ( y e. x -> ( ( H ` y ) = w -> ( ( w e. ( H " x ) /\ ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) -> ( x i^i ( `' R " { y } ) ) = (/) ) ) ) )
53	52	com3l 89	. . . . . . . . . . . 12 |- ( y e. x -> ( ( H ` y ) = w -> ( ( ph /\ x C_ A ) -> ( ( w e. ( H " x ) /\ ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) -> ( x i^i ( `' R " { y } ) ) = (/) ) ) ) )
54	53	com4t 93	. . . . . . . . . . 11 |- ( ( ph /\ x C_ A ) -> ( ( w e. ( H " x ) /\ ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) -> ( y e. x -> ( ( H ` y ) = w -> ( x i^i ( `' R " { y } ) ) = (/) ) ) ) )
55	54	imp 445	. . . . . . . . . 10 |- ( ( ( ph /\ x C_ A ) /\ ( w e. ( H " x ) /\ ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) ) -> ( y e. x -> ( ( H ` y ) = w -> ( x i^i ( `' R " { y } ) ) = (/) ) ) )
56	55	reximdvai 3015	. . . . . . . . 9 |- ( ( ( ph /\ x C_ A ) /\ ( w e. ( H " x ) /\ ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) ) -> ( E. y e. x ( H ` y ) = w -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) )
57	39, 56	mpd 15	. . . . . . . 8 |- ( ( ( ph /\ x C_ A ) /\ ( w e. ( H " x ) /\ ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) ) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) )
58	57	rexlimdvaa 3032	. . . . . . 7 |- ( ( ph /\ x C_ A ) -> ( E. w e. ( H " x ) ( ( H " x ) i^i ( `' S " { w } ) ) = (/) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) )
59	58	ex 450	. . . . . 6 |- ( ph -> ( x C_ A -> ( E. w e. ( H " x ) ( ( H " x ) i^i ( `' S " { w } ) ) = (/) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) ) )
60	59	adantrd 484	. . . . 5 |- ( ph -> ( ( x C_ A /\ x =/= (/) ) -> ( E. w e. ( H " x ) ( ( H " x ) i^i ( `' S " { w } ) ) = (/) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) ) )
61	60	a2d 29	. . . 4 |- ( ph -> ( ( ( x C_ A /\ x =/= (/) ) -> E. w e. ( H " x ) ( ( H " x ) i^i ( `' S " { w } ) ) = (/) ) -> ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) ) )
62	33, 61	syld 47	. . 3 |- ( ph -> ( S Fr B -> ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) ) )
63	62	alrimdv 1857	. 2 |- ( ph -> ( S Fr B -> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) ) )
64		dffr3 5498	. 2 |- ( R Fr A <-> A. x ( ( x C_ A /\ x =/= (/) ) -> E. y e. x ( x i^i ( `' R " { y } ) ) = (/) ) )
65	63, 64	syl6ibr 242	1 |- ( ph -> ( S Fr B -> R Fr A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   Fr wfr 5070   `'ccnv 5113   ran crn 5115   "cima 5117   Fun wfun 5882   Fn wfn 5883   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-fr 5073  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-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

This theorem is referenced by:  isofr  6592  isofr2  6594  isowe2  6600