Theorem wlkntrllem3 25291

Description: Lemma 3 for wlkntrl 25292: F is not injective. (Contributed by Alexander van der Vekens, 22-Oct-2017.)

Hypotheses

Ref	Expression
wlkntrl.v	|- V = { x , y }
wlkntrl.e	|- E = { <. 0 , { x , y } >. }
wlkntrl.f	|- F = { <. 0 , 0 >. , <. 1 , 0 >. }
wlkntrl.p	|- P = { <. 0 , x >. , <. 1 , y >. , <. 2 , x >. }

Assertion

Ref	Expression
wlkntrllem3	|- -. Fun `' F

Distinct variable group:   x, F, y

Allowed substitution hints:   P( x, y)   E( x, y)   V( x, y)

Proof of Theorem wlkntrllem3

Step	Hyp	Ref	Expression
1		ax-1ne0 9608	. . 3 |- 1 =/= 0
2	1	neii 2626	. 2 |- -. 1 = 0
3		0ne1 10677	. . . . 5 |- 0 =/= 1
4		c0ex 9637	. . . . . 6 |- 0 e. _V
5		1ex 9638	. . . . . 6 |- 1 e. _V
6	4, 5, 4, 4	fpr 6072	. . . . 5 |- ( 0 =/= 1 -> { <. 0 , 0 >. , <. 1 , 0 >. } : { 0 , 1 } --> { 0 , 0 } )
7	3, 6	ax-mp 5	. . . 4 |- { <. 0 , 0 >. , <. 1 , 0 >. } : { 0 , 1 } --> { 0 , 0 }
8		wlkntrl.f	. . . . . 6 |- F = { <. 0 , 0 >. , <. 1 , 0 >. }
9	8	eqcomi 2460	. . . . 5 |- { <. 0 , 0 >. , <. 1 , 0 >. } = F
10	9	feq1i 5720	. . . 4 |- ( { <. 0 , 0 >. , <. 1 , 0 >. } : { 0 , 1 } --> { 0 , 0 } <-> F : { 0 , 1 } --> { 0 , 0 } )
11	7, 10	mpbi 212	. . 3 |- F : { 0 , 1 } --> { 0 , 0 }
12		df-f1 5587	. . . 4 |- ( F : { 0 , 1 } -1-1-> { 0 , 0 } <-> ( F : { 0 , 1 } --> { 0 , 0 } /\ Fun `' F ) )
13		dff13 6159	. . . . 5 |- ( F : { 0 , 1 } -1-1-> { 0 , 0 } <-> ( F : { 0 , 1 } --> { 0 , 0 } /\ A. x e. { 0 , 1 } A. y e. { 0 , 1 } ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
14		fveq2 5865	. . . . . . . . . . 11 |- ( x = 0 -> ( F ` x ) = ( F ` 0 ) )
15	14	eqeq1d 2453	. . . . . . . . . 10 |- ( x = 0 -> ( ( F ` x ) = ( F ` y ) <-> ( F ` 0 ) = ( F ` y ) ) )
16		eqeq1 2455	. . . . . . . . . 10 |- ( x = 0 -> ( x = y <-> 0 = y ) )
17	15, 16	imbi12d 322	. . . . . . . . 9 |- ( x = 0 -> ( ( ( F ` x ) = ( F ` y ) -> x = y ) <-> ( ( F ` 0 ) = ( F ` y ) -> 0 = y ) ) )
18	17	ralbidv 2827	. . . . . . . 8 |- ( x = 0 -> ( A. y e. { 0 , 1 } ( ( F ` x ) = ( F ` y ) -> x = y ) <-> A. y e. { 0 , 1 } ( ( F ` 0 ) = ( F ` y ) -> 0 = y ) ) )
19		fveq2 5865	. . . . . . . . . . 11 |- ( x = 1 -> ( F ` x ) = ( F ` 1 ) )
20	19	eqeq1d 2453	. . . . . . . . . 10 |- ( x = 1 -> ( ( F ` x ) = ( F ` y ) <-> ( F ` 1 ) = ( F ` y ) ) )
21		eqeq1 2455	. . . . . . . . . 10 |- ( x = 1 -> ( x = y <-> 1 = y ) )
22	20, 21	imbi12d 322	. . . . . . . . 9 |- ( x = 1 -> ( ( ( F ` x ) = ( F ` y ) -> x = y ) <-> ( ( F ` 1 ) = ( F ` y ) -> 1 = y ) ) )
23	22	ralbidv 2827	. . . . . . . 8 |- ( x = 1 -> ( A. y e. { 0 , 1 } ( ( F ` x ) = ( F ` y ) -> x = y ) <-> A. y e. { 0 , 1 } ( ( F ` 1 ) = ( F ` y ) -> 1 = y ) ) )
24	4, 5, 18, 23	ralpr 4025	. . . . . . 7 |- ( A. x e. { 0 , 1 } A. y e. { 0 , 1 } ( ( F ` x ) = ( F ` y ) -> x = y ) <-> ( A. y e. { 0 , 1 } ( ( F ` 0 ) = ( F ` y ) -> 0 = y ) /\ A. y e. { 0 , 1 } ( ( F ` 1 ) = ( F ` y ) -> 1 = y ) ) )
25		fveq2 5865	. . . . . . . . . . 11 |- ( y = 0 -> ( F ` y ) = ( F ` 0 ) )
26	25	eqeq2d 2461	. . . . . . . . . 10 |- ( y = 0 -> ( ( F ` 0 ) = ( F ` y ) <-> ( F ` 0 ) = ( F ` 0 ) ) )
27		eqeq2 2462	. . . . . . . . . 10 |- ( y = 0 -> ( 0 = y <-> 0 = 0 ) )
28	26, 27	imbi12d 322	. . . . . . . . 9 |- ( y = 0 -> ( ( ( F ` 0 ) = ( F ` y ) -> 0 = y ) <-> ( ( F ` 0 ) = ( F ` 0 ) -> 0 = 0 ) ) )
29		fveq2 5865	. . . . . . . . . . 11 |- ( y = 1 -> ( F ` y ) = ( F ` 1 ) )
30	29	eqeq2d 2461	. . . . . . . . . 10 |- ( y = 1 -> ( ( F ` 0 ) = ( F ` y ) <-> ( F ` 0 ) = ( F ` 1 ) ) )
31		eqeq2 2462	. . . . . . . . . 10 |- ( y = 1 -> ( 0 = y <-> 0 = 1 ) )
32	30, 31	imbi12d 322	. . . . . . . . 9 |- ( y = 1 -> ( ( ( F ` 0 ) = ( F ` y ) -> 0 = y ) <-> ( ( F ` 0 ) = ( F ` 1 ) -> 0 = 1 ) ) )
33	4, 5, 28, 32	ralpr 4025	. . . . . . . 8 |- ( A. y e. { 0 , 1 } ( ( F ` 0 ) = ( F ` y ) -> 0 = y ) <-> ( ( ( F ` 0 ) = ( F ` 0 ) -> 0 = 0 ) /\ ( ( F ` 0 ) = ( F ` 1 ) -> 0 = 1 ) ) )
34	25	eqeq2d 2461	. . . . . . . . . 10 |- ( y = 0 -> ( ( F ` 1 ) = ( F ` y ) <-> ( F ` 1 ) = ( F ` 0 ) ) )
35		eqeq2 2462	. . . . . . . . . 10 |- ( y = 0 -> ( 1 = y <-> 1 = 0 ) )
36	34, 35	imbi12d 322	. . . . . . . . 9 |- ( y = 0 -> ( ( ( F ` 1 ) = ( F ` y ) -> 1 = y ) <-> ( ( F ` 1 ) = ( F ` 0 ) -> 1 = 0 ) ) )
37	29	eqeq2d 2461	. . . . . . . . . 10 |- ( y = 1 -> ( ( F ` 1 ) = ( F ` y ) <-> ( F ` 1 ) = ( F ` 1 ) ) )
38		eqeq2 2462	. . . . . . . . . 10 |- ( y = 1 -> ( 1 = y <-> 1 = 1 ) )
39	37, 38	imbi12d 322	. . . . . . . . 9 |- ( y = 1 -> ( ( ( F ` 1 ) = ( F ` y ) -> 1 = y ) <-> ( ( F ` 1 ) = ( F ` 1 ) -> 1 = 1 ) ) )
40	4, 5, 36, 39	ralpr 4025	. . . . . . . 8 |- ( A. y e. { 0 , 1 } ( ( F ` 1 ) = ( F ` y ) -> 1 = y ) <-> ( ( ( F ` 1 ) = ( F ` 0 ) -> 1 = 0 ) /\ ( ( F ` 1 ) = ( F ` 1 ) -> 1 = 1 ) ) )
41	8	fveq1i 5866	. . . . . . . . . . . . 13 |- ( F ` 1 ) = ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 1 )
42	5, 4	fvpr2 6108	. . . . . . . . . . . . . 14 |- ( 0 =/= 1 -> ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 1 ) = 0 )
43	3, 42	mp1i 13	. . . . . . . . . . . . 13 |- ( -. 1 = 0 -> ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 1 ) = 0 )
44	41, 43	syl5eq 2497	. . . . . . . . . . . 12 |- ( -. 1 = 0 -> ( F ` 1 ) = 0 )
45	8	fveq1i 5866	. . . . . . . . . . . . 13 |- ( F ` 0 ) = ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 0 )
46	4, 4	fvpr1 6107	. . . . . . . . . . . . . 14 |- ( 0 =/= 1 -> ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 0 ) = 0 )
47	3, 46	mp1i 13	. . . . . . . . . . . . 13 |- ( -. 1 = 0 -> ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 0 ) = 0 )
48	45, 47	syl5req 2498	. . . . . . . . . . . 12 |- ( -. 1 = 0 -> 0 = ( F ` 0 ) )
49	44, 48	eqtrd 2485	. . . . . . . . . . 11 |- ( -. 1 = 0 -> ( F ` 1 ) = ( F ` 0 ) )
50	49	con1i 133	. . . . . . . . . 10 |- ( -. ( F ` 1 ) = ( F ` 0 ) -> 1 = 0 )
51		id 22	. . . . . . . . . 10 |- ( 1 = 0 -> 1 = 0 )
52	50, 51	ja 165	. . . . . . . . 9 |- ( ( ( F ` 1 ) = ( F ` 0 ) -> 1 = 0 ) -> 1 = 0 )
53	52	ad2antrl 734	. . . . . . . 8 |- ( ( ( ( ( F ` 0 ) = ( F ` 0 ) -> 0 = 0 ) /\ ( ( F ` 0 ) = ( F ` 1 ) -> 0 = 1 ) ) /\ ( ( ( F ` 1 ) = ( F ` 0 ) -> 1 = 0 ) /\ ( ( F ` 1 ) = ( F ` 1 ) -> 1 = 1 ) ) ) -> 1 = 0 )
54	33, 40, 53	syl2anb 482	. . . . . . 7 |- ( ( A. y e. { 0 , 1 } ( ( F ` 0 ) = ( F ` y ) -> 0 = y ) /\ A. y e. { 0 , 1 } ( ( F ` 1 ) = ( F ` y ) -> 1 = y ) ) -> 1 = 0 )
55	24, 54	sylbi 199	. . . . . 6 |- ( A. x e. { 0 , 1 } A. y e. { 0 , 1 } ( ( F ` x ) = ( F ` y ) -> x = y ) -> 1 = 0 )
56	55	adantl 468	. . . . 5 |- ( ( F : { 0 , 1 } --> { 0 , 0 } /\ A. x e. { 0 , 1 } A. y e. { 0 , 1 } ( ( F ` x ) = ( F ` y ) -> x = y ) ) -> 1 = 0 )
57	13, 56	sylbi 199	. . . 4 |- ( F : { 0 , 1 } -1-1-> { 0 , 0 } -> 1 = 0 )
58	12, 57	sylbir 217	. . 3 |- ( ( F : { 0 , 1 } --> { 0 , 0 } /\ Fun `' F ) -> 1 = 0 )
59	11, 58	mpan 676	. 2 |- ( Fun `' F -> 1 = 0 )
60	2, 59	mto 180	1 |- -. Fun `' F

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1444   =/= wne 2622   A.wral 2737   {csn 3968   {cpr 3970   {ctp 3972   <.cop 3974   `'ccnv 4833   Fun wfun 5576   -->wf 5578   -1-1->wf1 5579   ` cfv 5582   0cc0 9539   1c1 9540   2c2 10659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-mulcl 9601  ax-i2m1 9607  ax-1ne0 9608

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fv 5590

This theorem is referenced by:  wlkntrl  25292