Theorem fnwe2lem3 37622

Description: Lemma for fnwe2 37623. Trichotomy. (Contributed by Stefan O'Rear, 19-Jan-2015.)

Hypotheses

Ref	Expression
fnwe2.su	|- ( z = ( F ` x ) -> S = U )
fnwe2.t	|- T = { <. x , y >. | ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x U y ) ) }
fnwe2.s	|- ( ( ph /\ x e. A ) -> U We { y e. A | ( F ` y ) = ( F ` x ) } )
fnwe2.f	|- ( ph -> ( F |` A ) : A --> B )
fnwe2.r	|- ( ph -> R We B )
fnwe2lem3.a	|- ( ph -> a e. A )
fnwe2lem3.b	|- ( ph -> b e. A )

Assertion

Ref	Expression
fnwe2lem3	|- ( ph -> ( a T b \/ a = b \/ b T a ) )

Distinct variable groups:   y, U, z, a, b   x, S, y, a, b   x, R, y, a, b   ph, x, y, z   x, A, y, z, a, b   x, F, y, z, a, b   T, a, b   B, a, b

Allowed substitution hints:   ph( a, b)   B( x, y, z)   R( z)   S( z)   T( x, y, z)   U( x)

Proof of Theorem fnwe2lem3

Step	Hyp	Ref	Expression
1		orc 400	. . . . 5 |- ( ( F ` a ) R ( F ` b ) -> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) )
2	1	adantl 482	. . . 4 |- ( ( ph /\ ( F ` a ) R ( F ` b ) ) -> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) )
3		fnwe2.su	. . . . 5 |- ( z = ( F ` x ) -> S = U )
4		fnwe2.t	. . . . 5 |- T = { <. x , y >. | ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x U y ) ) }
5	3, 4	fnwe2val 37619	. . . 4 |- ( a T b <-> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) )
6	2, 5	sylibr 224	. . 3 |- ( ( ph /\ ( F ` a ) R ( F ` b ) ) -> a T b )
7	6	3mix1d 1236	. 2 |- ( ( ph /\ ( F ` a ) R ( F ` b ) ) -> ( a T b \/ a = b \/ b T a ) )
8		simplr 792	. . . . . . 7 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> ( F ` a ) = ( F ` b ) )
9		simpr 477	. . . . . . 7 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> a [_ ( F ` a ) / z ]_ S b )
10	8, 9	jca 554	. . . . . 6 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) )
11	10	olcd 408	. . . . 5 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> ( ( F ` a ) R ( F ` b ) \/ ( ( F ` a ) = ( F ` b ) /\ a [_ ( F ` a ) / z ]_ S b ) ) )
12	11, 5	sylibr 224	. . . 4 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> a T b )
13	12	3mix1d 1236	. . 3 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a [_ ( F ` a ) / z ]_ S b ) -> ( a T b \/ a = b \/ b T a ) )
14		3mix2 1231	. . . 4 |- ( a = b -> ( a T b \/ a = b \/ b T a ) )
15	14	adantl 482	. . 3 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ a = b ) -> ( a T b \/ a = b \/ b T a ) )
16		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> ( F ` a ) = ( F ` b ) )
17	16	eqcomd 2628	. . . . . . 7 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> ( F ` b ) = ( F ` a ) )
18		csbeq1 3536	. . . . . . . . . 10 |- ( ( F ` a ) = ( F ` b ) -> [_ ( F ` a ) / z ]_ S = [_ ( F ` b ) / z ]_ S )
19	18	adantl 482	. . . . . . . . 9 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> [_ ( F ` a ) / z ]_ S = [_ ( F ` b ) / z ]_ S )
20	19	breqd 4664	. . . . . . . 8 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( b [_ ( F ` a ) / z ]_ S a <-> b [_ ( F ` b ) / z ]_ S a ) )
21	20	biimpa 501	. . . . . . 7 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> b [_ ( F ` b ) / z ]_ S a )
22	17, 21	jca 554	. . . . . 6 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> ( ( F ` b ) = ( F ` a ) /\ b [_ ( F ` b ) / z ]_ S a ) )
23	22	olcd 408	. . . . 5 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> ( ( F ` b ) R ( F ` a ) \/ ( ( F ` b ) = ( F ` a ) /\ b [_ ( F ` b ) / z ]_ S a ) ) )
24	3, 4	fnwe2val 37619	. . . . 5 |- ( b T a <-> ( ( F ` b ) R ( F ` a ) \/ ( ( F ` b ) = ( F ` a ) /\ b [_ ( F ` b ) / z ]_ S a ) ) )
25	23, 24	sylibr 224	. . . 4 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> b T a )
26	25	3mix3d 1238	. . 3 |- ( ( ( ph /\ ( F ` a ) = ( F ` b ) ) /\ b [_ ( F ` a ) / z ]_ S a ) -> ( a T b \/ a = b \/ b T a ) )
27		fnwe2lem3.a	. . . . . . 7 |- ( ph -> a e. A )
28		fnwe2.s	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> U We { y e. A | ( F ` y ) = ( F ` x ) } )
29	3, 4, 28	fnwe2lem1 37620	. . . . . . 7 |- ( ( ph /\ a e. A ) -> [_ ( F ` a ) / z ]_ S We { y e. A | ( F ` y ) = ( F ` a ) } )
30	27, 29	mpdan 702	. . . . . 6 |- ( ph -> [_ ( F ` a ) / z ]_ S We { y e. A | ( F ` y ) = ( F ` a ) } )
31		weso 5105	. . . . . 6 |- ( [_ ( F ` a ) / z ]_ S We { y e. A | ( F ` y ) = ( F ` a ) } -> [_ ( F ` a ) / z ]_ S Or { y e. A | ( F ` y ) = ( F ` a ) } )
32	30, 31	syl 17	. . . . 5 |- ( ph -> [_ ( F ` a ) / z ]_ S Or { y e. A | ( F ` y ) = ( F ` a ) } )
33	32	adantr 481	. . . 4 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> [_ ( F ` a ) / z ]_ S Or { y e. A | ( F ` y ) = ( F ` a ) } )
34	27	adantr 481	. . . . 5 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> a e. A )
35		eqidd 2623	. . . . 5 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( F ` a ) = ( F ` a ) )
36		fveq2 6191	. . . . . . 7 |- ( y = a -> ( F ` y ) = ( F ` a ) )
37	36	eqeq1d 2624	. . . . . 6 |- ( y = a -> ( ( F ` y ) = ( F ` a ) <-> ( F ` a ) = ( F ` a ) ) )
38	37	elrab 3363	. . . . 5 |- ( a e. { y e. A | ( F ` y ) = ( F ` a ) } <-> ( a e. A /\ ( F ` a ) = ( F ` a ) ) )
39	34, 35, 38	sylanbrc 698	. . . 4 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> a e. { y e. A | ( F ` y ) = ( F ` a ) } )
40		fnwe2lem3.b	. . . . . 6 |- ( ph -> b e. A )
41	40	adantr 481	. . . . 5 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> b e. A )
42		simpr 477	. . . . . 6 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( F ` a ) = ( F ` b ) )
43	42	eqcomd 2628	. . . . 5 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( F ` b ) = ( F ` a ) )
44		fveq2 6191	. . . . . . 7 |- ( y = b -> ( F ` y ) = ( F ` b ) )
45	44	eqeq1d 2624	. . . . . 6 |- ( y = b -> ( ( F ` y ) = ( F ` a ) <-> ( F ` b ) = ( F ` a ) ) )
46	45	elrab 3363	. . . . 5 |- ( b e. { y e. A | ( F ` y ) = ( F ` a ) } <-> ( b e. A /\ ( F ` b ) = ( F ` a ) ) )
47	41, 43, 46	sylanbrc 698	. . . 4 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> b e. { y e. A | ( F ` y ) = ( F ` a ) } )
48		solin 5058	. . . 4 |- ( ( [_ ( F ` a ) / z ]_ S Or { y e. A | ( F ` y ) = ( F ` a ) } /\ ( a e. { y e. A | ( F ` y ) = ( F ` a ) } /\ b e. { y e. A | ( F ` y ) = ( F ` a ) } ) ) -> ( a [_ ( F ` a ) / z ]_ S b \/ a = b \/ b [_ ( F ` a ) / z ]_ S a ) )
49	33, 39, 47, 48	syl12anc 1324	. . 3 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( a [_ ( F ` a ) / z ]_ S b \/ a = b \/ b [_ ( F ` a ) / z ]_ S a ) )
50	13, 15, 26, 49	mpjao3dan 1395	. 2 |- ( ( ph /\ ( F ` a ) = ( F ` b ) ) -> ( a T b \/ a = b \/ b T a ) )
51		orc 400	. . . . 5 |- ( ( F ` b ) R ( F ` a ) -> ( ( F ` b ) R ( F ` a ) \/ ( ( F ` b ) = ( F ` a ) /\ b [_ ( F ` b ) / z ]_ S a ) ) )
52	51	adantl 482	. . . 4 |- ( ( ph /\ ( F ` b ) R ( F ` a ) ) -> ( ( F ` b ) R ( F ` a ) \/ ( ( F ` b ) = ( F ` a ) /\ b [_ ( F ` b ) / z ]_ S a ) ) )
53	52, 24	sylibr 224	. . 3 |- ( ( ph /\ ( F ` b ) R ( F ` a ) ) -> b T a )
54	53	3mix3d 1238	. 2 |- ( ( ph /\ ( F ` b ) R ( F ` a ) ) -> ( a T b \/ a = b \/ b T a ) )
55		fnwe2.r	. . . 4 |- ( ph -> R We B )
56		weso 5105	. . . 4 |- ( R We B -> R Or B )
57	55, 56	syl 17	. . 3 |- ( ph -> R Or B )
58		fvres 6207	. . . . 5 |- ( a e. A -> ( ( F |` A ) ` a ) = ( F ` a ) )
59	27, 58	syl 17	. . . 4 |- ( ph -> ( ( F |` A ) ` a ) = ( F ` a ) )
60		fnwe2.f	. . . . 5 |- ( ph -> ( F |` A ) : A --> B )
61	60, 27	ffvelrnd 6360	. . . 4 |- ( ph -> ( ( F |` A ) ` a ) e. B )
62	59, 61	eqeltrrd 2702	. . 3 |- ( ph -> ( F ` a ) e. B )
63		fvres 6207	. . . . 5 |- ( b e. A -> ( ( F |` A ) ` b ) = ( F ` b ) )
64	40, 63	syl 17	. . . 4 |- ( ph -> ( ( F |` A ) ` b ) = ( F ` b ) )
65	60, 40	ffvelrnd 6360	. . . 4 |- ( ph -> ( ( F |` A ) ` b ) e. B )
66	64, 65	eqeltrrd 2702	. . 3 |- ( ph -> ( F ` b ) e. B )
67		solin 5058	. . 3 |- ( ( R Or B /\ ( ( F ` a ) e. B /\ ( F ` b ) e. B ) ) -> ( ( F ` a ) R ( F ` b ) \/ ( F ` a ) = ( F ` b ) \/ ( F ` b ) R ( F ` a ) ) )
68	57, 62, 66, 67	syl12anc 1324	. 2 |- ( ph -> ( ( F ` a ) R ( F ` b ) \/ ( F ` a ) = ( F ` b ) \/ ( F ` b ) R ( F ` a ) ) )
69	7, 50, 54, 68	mpjao3dan 1395	1 |- ( ph -> ( a T b \/ a = b \/ b T a ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   {crab 2916   [_csb 3533   class class class wbr 4653   {copab 4712   Or wor 5034   We wwe 5072   |` cres 5116   -->wf 5884   ` cfv 5888

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-csb 3534  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-fr 5073  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  fnwe2  37623