Theorem wlkntrllem3 24689

Description: Lemma 3 for wlkntrl 24690: 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 9578	. . 3 |- 1 =/= 0
2	1	neii 2656	. 2 |- -. 1 = 0
3		0ne1 10624	. . . . 5 |- 0 =/= 1
4		c0ex 9607	. . . . . 6 |- 0 e. _V
5		1ex 9608	. . . . . 6 |- 1 e. _V
6	4, 5, 4, 4	fpr 6080	. . . . 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 2470	. . . . 5 |- { <. 0 , 0 >. , <. 1 , 0 >. } = F
10	9	feq1i 5729	. . . 4 |- ( { <. 0 , 0 >. , <. 1 , 0 >. } : { 0 , 1 } --> { 0 , 0 } <-> F : { 0 , 1 } --> { 0 , 0 } )
11	7, 10	mpbi 208	. . 3 |- F : { 0 , 1 } --> { 0 , 0 }
12		df-f1 5599	. . . 4 |- ( F : { 0 , 1 } -1-1-> { 0 , 0 } <-> ( F : { 0 , 1 } --> { 0 , 0 } /\ Fun `' F ) )
13		dff13 6167	. . . . 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 5872	. . . . . . . . . . 11 |- ( x = 0 -> ( F ` x ) = ( F ` 0 ) )
15	14	eqeq1d 2459	. . . . . . . . . 10 |- ( x = 0 -> ( ( F ` x ) = ( F ` y ) <-> ( F ` 0 ) = ( F ` y ) ) )
16		eqeq1 2461	. . . . . . . . . 10 |- ( x = 0 -> ( x = y <-> 0 = y ) )
17	15, 16	imbi12d 320	. . . . . . . . 9 |- ( x = 0 -> ( ( ( F ` x ) = ( F ` y ) -> x = y ) <-> ( ( F ` 0 ) = ( F ` y ) -> 0 = y ) ) )
18	17	ralbidv 2896	. . . . . . . 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 5872	. . . . . . . . . . 11 |- ( x = 1 -> ( F ` x ) = ( F ` 1 ) )
20	19	eqeq1d 2459	. . . . . . . . . 10 |- ( x = 1 -> ( ( F ` x ) = ( F ` y ) <-> ( F ` 1 ) = ( F ` y ) ) )
21		eqeq1 2461	. . . . . . . . . 10 |- ( x = 1 -> ( x = y <-> 1 = y ) )
22	20, 21	imbi12d 320	. . . . . . . . 9 |- ( x = 1 -> ( ( ( F ` x ) = ( F ` y ) -> x = y ) <-> ( ( F ` 1 ) = ( F ` y ) -> 1 = y ) ) )
23	22	ralbidv 2896	. . . . . . . 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 4085	. . . . . . 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 5872	. . . . . . . . . . 11 |- ( y = 0 -> ( F ` y ) = ( F ` 0 ) )
26	25	eqeq2d 2471	. . . . . . . . . 10 |- ( y = 0 -> ( ( F ` 0 ) = ( F ` y ) <-> ( F ` 0 ) = ( F ` 0 ) ) )
27		eqeq2 2472	. . . . . . . . . 10 |- ( y = 0 -> ( 0 = y <-> 0 = 0 ) )
28	26, 27	imbi12d 320	. . . . . . . . 9 |- ( y = 0 -> ( ( ( F ` 0 ) = ( F ` y ) -> 0 = y ) <-> ( ( F ` 0 ) = ( F ` 0 ) -> 0 = 0 ) ) )
29		fveq2 5872	. . . . . . . . . . 11 |- ( y = 1 -> ( F ` y ) = ( F ` 1 ) )
30	29	eqeq2d 2471	. . . . . . . . . 10 |- ( y = 1 -> ( ( F ` 0 ) = ( F ` y ) <-> ( F ` 0 ) = ( F ` 1 ) ) )
31		eqeq2 2472	. . . . . . . . . 10 |- ( y = 1 -> ( 0 = y <-> 0 = 1 ) )
32	30, 31	imbi12d 320	. . . . . . . . 9 |- ( y = 1 -> ( ( ( F ` 0 ) = ( F ` y ) -> 0 = y ) <-> ( ( F ` 0 ) = ( F ` 1 ) -> 0 = 1 ) ) )
33	4, 5, 28, 32	ralpr 4085	. . . . . . . 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 2471	. . . . . . . . . 10 |- ( y = 0 -> ( ( F ` 1 ) = ( F ` y ) <-> ( F ` 1 ) = ( F ` 0 ) ) )
35		eqeq2 2472	. . . . . . . . . 10 |- ( y = 0 -> ( 1 = y <-> 1 = 0 ) )
36	34, 35	imbi12d 320	. . . . . . . . 9 |- ( y = 0 -> ( ( ( F ` 1 ) = ( F ` y ) -> 1 = y ) <-> ( ( F ` 1 ) = ( F ` 0 ) -> 1 = 0 ) ) )
37	29	eqeq2d 2471	. . . . . . . . . 10 |- ( y = 1 -> ( ( F ` 1 ) = ( F ` y ) <-> ( F ` 1 ) = ( F ` 1 ) ) )
38		eqeq2 2472	. . . . . . . . . 10 |- ( y = 1 -> ( 1 = y <-> 1 = 1 ) )
39	37, 38	imbi12d 320	. . . . . . . . 9 |- ( y = 1 -> ( ( ( F ` 1 ) = ( F ` y ) -> 1 = y ) <-> ( ( F ` 1 ) = ( F ` 1 ) -> 1 = 1 ) ) )
40	4, 5, 36, 39	ralpr 4085	. . . . . . . 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 5873	. . . . . . . . . . . . 13 |- ( F ` 1 ) = ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 1 )
42	5, 4	fvpr2 6116	. . . . . . . . . . . . . 14 |- ( 0 =/= 1 -> ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 1 ) = 0 )
43	3, 42	mp1i 12	. . . . . . . . . . . . 13 |- ( -. 1 = 0 -> ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 1 ) = 0 )
44	41, 43	syl5eq 2510	. . . . . . . . . . . 12 |- ( -. 1 = 0 -> ( F ` 1 ) = 0 )
45	8	fveq1i 5873	. . . . . . . . . . . . 13 |- ( F ` 0 ) = ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 0 )
46	4, 4	fvpr1 6115	. . . . . . . . . . . . . 14 |- ( 0 =/= 1 -> ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 0 ) = 0 )
47	3, 46	mp1i 12	. . . . . . . . . . . . 13 |- ( -. 1 = 0 -> ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 0 ) = 0 )
48	45, 47	syl5req 2511	. . . . . . . . . . . 12 |- ( -. 1 = 0 -> 0 = ( F ` 0 ) )
49	44, 48	eqtrd 2498	. . . . . . . . . . 11 |- ( -. 1 = 0 -> ( F ` 1 ) = ( F ` 0 ) )
50	49	con1i 129	. . . . . . . . . 10 |- ( -. ( F ` 1 ) = ( F ` 0 ) -> 1 = 0 )
51		id 22	. . . . . . . . . 10 |- ( 1 = 0 -> 1 = 0 )
52	50, 51	ja 161	. . . . . . . . 9 |- ( ( ( F ` 1 ) = ( F ` 0 ) -> 1 = 0 ) -> 1 = 0 )
53	52	ad2antrl 727	. . . . . . . 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 479	. . . . . . 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 195	. . . . . 6 |- ( A. x e. { 0 , 1 } A. y e. { 0 , 1 } ( ( F ` x ) = ( F ` y ) -> x = y ) -> 1 = 0 )
56	55	adantl 466	. . . . 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 195	. . . 4 |- ( F : { 0 , 1 } -1-1-> { 0 , 0 } -> 1 = 0 )
58	12, 57	sylbir 213	. . 3 |- ( ( F : { 0 , 1 } --> { 0 , 0 } /\ Fun `' F ) -> 1 = 0 )
59	11, 58	mpan 670	. 2 |- ( Fun `' F -> 1 = 0 )
60	2, 59	mto 176	1 |- -. Fun `' F

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1395   =/= wne 2652   A.wral 2807   {csn 4032   {cpr 4034   {ctp 4036   <.cop 4038   `'ccnv 5007   Fun wfun 5588   -->wf 5590   -1-1->wf1 5591   ` cfv 5594   0cc0 9509   1c1 9510   2c2 10606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-mulcl 9571  ax-i2m1 9577  ax-1ne0 9578

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fv 5602

This theorem is referenced by:  wlkntrl  24690