Theorem wlkntrllem3 23605

Description: Lemma 3 for wlkntrl 23606: 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 9455	. . 3 |- 1 =/= 0
2	1	neii 2648	. 2 |- -. 1 = 0
3		0ne1 10493	. . . . 5 |- 0 =/= 1
4		c0ex 9484	. . . . . 6 |- 0 e. _V
5		1ex 9485	. . . . . 6 |- 1 e. _V
6	4, 5, 4, 4	fpr 5992	. . . . 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 2464	. . . . 5 |- { <. 0 , 0 >. , <. 1 , 0 >. } = F
10	9	feq1i 5652	. . . 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 5524	. . . 4 |- ( F : { 0 , 1 } -1-1-> { 0 , 0 } <-> ( F : { 0 , 1 } --> { 0 , 0 } /\ Fun `' F ) )
13		dff13 6073	. . . . 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 5792	. . . . . . . . . . 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 320	. . . . . . . . 9 |- ( x = 0 -> ( ( ( F ` x ) = ( F ` y ) -> x = y ) <-> ( ( F ` 0 ) = ( F ` y ) -> 0 = y ) ) )
18	17	ralbidv 2841	. . . . . . . 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 5792	. . . . . . . . . . 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 320	. . . . . . . . 9 |- ( x = 1 -> ( ( ( F ` x ) = ( F ` y ) -> x = y ) <-> ( ( F ` 1 ) = ( F ` y ) -> 1 = y ) ) )
23	22	ralbidv 2841	. . . . . . . 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 4030	. . . . . . 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 5792	. . . . . . . . . . 11 |- ( y = 0 -> ( F ` y ) = ( F ` 0 ) )
26	25	eqeq2d 2465	. . . . . . . . . 10 |- ( y = 0 -> ( ( F ` 0 ) = ( F ` y ) <-> ( F ` 0 ) = ( F ` 0 ) ) )
27		eqeq2 2466	. . . . . . . . . 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 5792	. . . . . . . . . . 11 |- ( y = 1 -> ( F ` y ) = ( F ` 1 ) )
30	29	eqeq2d 2465	. . . . . . . . . 10 |- ( y = 1 -> ( ( F ` 0 ) = ( F ` y ) <-> ( F ` 0 ) = ( F ` 1 ) ) )
31		eqeq2 2466	. . . . . . . . . 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 4030	. . . . . . . 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 2465	. . . . . . . . . 10 |- ( y = 0 -> ( ( F ` 1 ) = ( F ` y ) <-> ( F ` 1 ) = ( F ` 0 ) ) )
35		eqeq2 2466	. . . . . . . . . 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 2465	. . . . . . . . . 10 |- ( y = 1 -> ( ( F ` 1 ) = ( F ` y ) <-> ( F ` 1 ) = ( F ` 1 ) ) )
38		eqeq2 2466	. . . . . . . . . 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 4030	. . . . . . . 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 5793	. . . . . . . . . . . . 13 |- ( F ` 1 ) = ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 1 )
42	5, 4	fvpr2 6024	. . . . . . . . . . . . . 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 2504	. . . . . . . . . . . 12 |- ( -. 1 = 0 -> ( F ` 1 ) = 0 )
45	8	fveq1i 5793	. . . . . . . . . . . . 13 |- ( F ` 0 ) = ( { <. 0 , 0 >. , <. 1 , 0 >. } ` 0 )
46	4, 4	fvpr1 6023	. . . . . . . . . . . . . 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 2505	. . . . . . . . . . . 12 |- ( -. 1 = 0 -> 0 = ( F ` 0 ) )
49	44, 48	eqtrd 2492	. . . . . . . . . . 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 1370   =/= wne 2644   A.wral 2795   {csn 3978   {cpr 3980   {ctp 3982   <.cop 3984   `'ccnv 4940   Fun wfun 5513   -->wf 5515   -1-1->wf1 5516   ` cfv 5519   0cc0 9386   1c1 9387   2c2 10475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-mulcl 9448  ax-i2m1 9454  ax-1ne0 9455

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fv 5527

This theorem is referenced by:  wlkntrl  23606