Theorem isosolem 6597

Description: Lemma for isoso 6598. (Contributed by Stefan O'Rear, 16-Nov-2014.)

Assertion

Ref	Expression
isosolem	|- ( H Isom R , S ( A , B ) -> ( S Or B -> R Or A ) )

Proof of Theorem isosolem

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

Step	Hyp	Ref	Expression
1		isopolem 6595	. . 3 |- ( H Isom R , S ( A , B ) -> ( S Po B -> R Po A ) )
2		isof1o 6573	. . . . . . . 8 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
3		f1of 6137	. . . . . . . 8 |- ( H : A -1-1-onto-> B -> H : A --> B )
4		ffvelrn 6357	. . . . . . . . . 10 |- ( ( H : A --> B /\ c e. A ) -> ( H ` c ) e. B )
5	4	ex 450	. . . . . . . . 9 |- ( H : A --> B -> ( c e. A -> ( H ` c ) e. B ) )
6		ffvelrn 6357	. . . . . . . . . 10 |- ( ( H : A --> B /\ d e. A ) -> ( H ` d ) e. B )
7	6	ex 450	. . . . . . . . 9 |- ( H : A --> B -> ( d e. A -> ( H ` d ) e. B ) )
8	5, 7	anim12d 586	. . . . . . . 8 |- ( H : A --> B -> ( ( c e. A /\ d e. A ) -> ( ( H ` c ) e. B /\ ( H ` d ) e. B ) ) )
9	2, 3, 8	3syl 18	. . . . . . 7 |- ( H Isom R , S ( A , B ) -> ( ( c e. A /\ d e. A ) -> ( ( H ` c ) e. B /\ ( H ` d ) e. B ) ) )
10	9	imp 445	. . . . . 6 |- ( ( H Isom R , S ( A , B ) /\ ( c e. A /\ d e. A ) ) -> ( ( H ` c ) e. B /\ ( H ` d ) e. B ) )
11		breq1 4656	. . . . . . . 8 |- ( a = ( H ` c ) -> ( a S b <-> ( H ` c ) S b ) )
12		eqeq1 2626	. . . . . . . 8 |- ( a = ( H ` c ) -> ( a = b <-> ( H ` c ) = b ) )
13		breq2 4657	. . . . . . . 8 |- ( a = ( H ` c ) -> ( b S a <-> b S ( H ` c ) ) )
14	11, 12, 13	3orbi123d 1398	. . . . . . 7 |- ( a = ( H ` c ) -> ( ( a S b \/ a = b \/ b S a ) <-> ( ( H ` c ) S b \/ ( H ` c ) = b \/ b S ( H ` c ) ) ) )
15		breq2 4657	. . . . . . . 8 |- ( b = ( H ` d ) -> ( ( H ` c ) S b <-> ( H ` c ) S ( H ` d ) ) )
16		eqeq2 2633	. . . . . . . 8 |- ( b = ( H ` d ) -> ( ( H ` c ) = b <-> ( H ` c ) = ( H ` d ) ) )
17		breq1 4656	. . . . . . . 8 |- ( b = ( H ` d ) -> ( b S ( H ` c ) <-> ( H ` d ) S ( H ` c ) ) )
18	15, 16, 17	3orbi123d 1398	. . . . . . 7 |- ( b = ( H ` d ) -> ( ( ( H ` c ) S b \/ ( H ` c ) = b \/ b S ( H ` c ) ) <-> ( ( H ` c ) S ( H ` d ) \/ ( H ` c ) = ( H ` d ) \/ ( H ` d ) S ( H ` c ) ) ) )
19	14, 18	rspc2v 3322	. . . . . 6 |- ( ( ( H ` c ) e. B /\ ( H ` d ) e. B ) -> ( A. a e. B A. b e. B ( a S b \/ a = b \/ b S a ) -> ( ( H ` c ) S ( H ` d ) \/ ( H ` c ) = ( H ` d ) \/ ( H ` d ) S ( H ` c ) ) ) )
20	10, 19	syl 17	. . . . 5 |- ( ( H Isom R , S ( A , B ) /\ ( c e. A /\ d e. A ) ) -> ( A. a e. B A. b e. B ( a S b \/ a = b \/ b S a ) -> ( ( H ` c ) S ( H ` d ) \/ ( H ` c ) = ( H ` d ) \/ ( H ` d ) S ( H ` c ) ) ) )
21		isorel 6576	. . . . . 6 |- ( ( H Isom R , S ( A , B ) /\ ( c e. A /\ d e. A ) ) -> ( c R d <-> ( H ` c ) S ( H ` d ) ) )
22		f1of1 6136	. . . . . . . . 9 |- ( H : A -1-1-onto-> B -> H : A -1-1-> B )
23	2, 22	syl 17	. . . . . . . 8 |- ( H Isom R , S ( A , B ) -> H : A -1-1-> B )
24		f1fveq 6519	. . . . . . . 8 |- ( ( H : A -1-1-> B /\ ( c e. A /\ d e. A ) ) -> ( ( H ` c ) = ( H ` d ) <-> c = d ) )
25	23, 24	sylan 488	. . . . . . 7 |- ( ( H Isom R , S ( A , B ) /\ ( c e. A /\ d e. A ) ) -> ( ( H ` c ) = ( H ` d ) <-> c = d ) )
26	25	bicomd 213	. . . . . 6 |- ( ( H Isom R , S ( A , B ) /\ ( c e. A /\ d e. A ) ) -> ( c = d <-> ( H ` c ) = ( H ` d ) ) )
27		isorel 6576	. . . . . . 7 |- ( ( H Isom R , S ( A , B ) /\ ( d e. A /\ c e. A ) ) -> ( d R c <-> ( H ` d ) S ( H ` c ) ) )
28	27	ancom2s 844	. . . . . 6 |- ( ( H Isom R , S ( A , B ) /\ ( c e. A /\ d e. A ) ) -> ( d R c <-> ( H ` d ) S ( H ` c ) ) )
29	21, 26, 28	3orbi123d 1398	. . . . 5 |- ( ( H Isom R , S ( A , B ) /\ ( c e. A /\ d e. A ) ) -> ( ( c R d \/ c = d \/ d R c ) <-> ( ( H ` c ) S ( H ` d ) \/ ( H ` c ) = ( H ` d ) \/ ( H ` d ) S ( H ` c ) ) ) )
30	20, 29	sylibrd 249	. . . 4 |- ( ( H Isom R , S ( A , B ) /\ ( c e. A /\ d e. A ) ) -> ( A. a e. B A. b e. B ( a S b \/ a = b \/ b S a ) -> ( c R d \/ c = d \/ d R c ) ) )
31	30	ralrimdvva 2974	. . 3 |- ( H Isom R , S ( A , B ) -> ( A. a e. B A. b e. B ( a S b \/ a = b \/ b S a ) -> A. c e. A A. d e. A ( c R d \/ c = d \/ d R c ) ) )
32	1, 31	anim12d 586	. 2 |- ( H Isom R , S ( A , B ) -> ( ( S Po B /\ A. a e. B A. b e. B ( a S b \/ a = b \/ b S a ) ) -> ( R Po A /\ A. c e. A A. d e. A ( c R d \/ c = d \/ d R c ) ) ) )
33		df-so 5036	. 2 |- ( S Or B <-> ( S Po B /\ A. a e. B A. b e. B ( a S b \/ a = b \/ b S a ) ) )
34		df-so 5036	. 2 |- ( R Or A <-> ( R Po A /\ A. c e. A A. d e. A ( c R d \/ c = d \/ d R c ) ) )
35	32, 33, 34	3imtr4g 285	1 |- ( H Isom R , S ( A , B ) -> ( S Or B -> R Or A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   Po wpo 5033   Or wor 5034   -->wf 5884   -1-1->wf1 5885   -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-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-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-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-f1o 5895  df-fv 5896  df-isom 5897

This theorem is referenced by:  isoso  6598  isowe2  6600