Theorem f1dom3fv3dif 6186

Description: The function values for a 1-1 function from a set with three different elements are different. (Contributed by AV, 20-Mar-2019.)

Hypotheses

Ref	Expression
f1dom3fv3dif.v	|- ( ph -> ( A e. X /\ B e. Y /\ C e. Z ) )
f1dom3fv3dif.n	|- ( ph -> ( A =/= B /\ A =/= C /\ B =/= C ) )
f1dom3fv3dif.f	|- ( ph -> F : { A , B , C } -1-1-> R )

Assertion

Ref	Expression
f1dom3fv3dif	|- ( ph -> ( ( F ` A ) =/= ( F ` B ) /\ ( F ` A ) =/= ( F ` C ) /\ ( F ` B ) =/= ( F ` C ) ) )

Proof of Theorem f1dom3fv3dif

Step	Hyp	Ref	Expression
1		f1dom3fv3dif.n	. . . 4 |- ( ph -> ( A =/= B /\ A =/= C /\ B =/= C ) )
2	1	simp1d 1042	. . 3 |- ( ph -> A =/= B )
3		f1dom3fv3dif.f	. . . . 5 |- ( ph -> F : { A , B , C } -1-1-> R )
4		eqidd 2472	. . . . . . 7 |- ( ph -> A = A )
5	4	3mix1d 1205	. . . . . 6 |- ( ph -> ( A = A \/ A = B \/ A = C ) )
6		f1dom3fv3dif.v	. . . . . . . 8 |- ( ph -> ( A e. X /\ B e. Y /\ C e. Z ) )
7	6	simp1d 1042	. . . . . . 7 |- ( ph -> A e. X )
8		eltpg 4005	. . . . . . 7 |- ( A e. X -> ( A e. { A , B , C } <-> ( A = A \/ A = B \/ A = C ) ) )
9	7, 8	syl 17	. . . . . 6 |- ( ph -> ( A e. { A , B , C } <-> ( A = A \/ A = B \/ A = C ) ) )
10	5, 9	mpbird 240	. . . . 5 |- ( ph -> A e. { A , B , C } )
11		eqidd 2472	. . . . . . 7 |- ( ph -> B = B )
12	11	3mix2d 1206	. . . . . 6 |- ( ph -> ( B = A \/ B = B \/ B = C ) )
13	6	simp2d 1043	. . . . . . 7 |- ( ph -> B e. Y )
14		eltpg 4005	. . . . . . 7 |- ( B e. Y -> ( B e. { A , B , C } <-> ( B = A \/ B = B \/ B = C ) ) )
15	13, 14	syl 17	. . . . . 6 |- ( ph -> ( B e. { A , B , C } <-> ( B = A \/ B = B \/ B = C ) ) )
16	12, 15	mpbird 240	. . . . 5 |- ( ph -> B e. { A , B , C } )
17		f1fveq 6181	. . . . 5 |- ( ( F : { A , B , C } -1-1-> R /\ ( A e. { A , B , C } /\ B e. { A , B , C } ) ) -> ( ( F ` A ) = ( F ` B ) <-> A = B ) )
18	3, 10, 16, 17	syl12anc 1290	. . . 4 |- ( ph -> ( ( F ` A ) = ( F ` B ) <-> A = B ) )
19	18	necon3bid 2687	. . 3 |- ( ph -> ( ( F ` A ) =/= ( F ` B ) <-> A =/= B ) )
20	2, 19	mpbird 240	. 2 |- ( ph -> ( F ` A ) =/= ( F ` B ) )
21	1	simp2d 1043	. . 3 |- ( ph -> A =/= C )
22	6	simp3d 1044	. . . . . 6 |- ( ph -> C e. Z )
23		tpid3g 4078	. . . . . 6 |- ( C e. Z -> C e. { A , B , C } )
24	22, 23	syl 17	. . . . 5 |- ( ph -> C e. { A , B , C } )
25		f1fveq 6181	. . . . 5 |- ( ( F : { A , B , C } -1-1-> R /\ ( A e. { A , B , C } /\ C e. { A , B , C } ) ) -> ( ( F ` A ) = ( F ` C ) <-> A = C ) )
26	3, 10, 24, 25	syl12anc 1290	. . . 4 |- ( ph -> ( ( F ` A ) = ( F ` C ) <-> A = C ) )
27	26	necon3bid 2687	. . 3 |- ( ph -> ( ( F ` A ) =/= ( F ` C ) <-> A =/= C ) )
28	21, 27	mpbird 240	. 2 |- ( ph -> ( F ` A ) =/= ( F ` C ) )
29	1	simp3d 1044	. . 3 |- ( ph -> B =/= C )
30		f1fveq 6181	. . . . 5 |- ( ( F : { A , B , C } -1-1-> R /\ ( B e. { A , B , C } /\ C e. { A , B , C } ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
31	3, 16, 24, 30	syl12anc 1290	. . . 4 |- ( ph -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
32	31	necon3bid 2687	. . 3 |- ( ph -> ( ( F ` B ) =/= ( F ` C ) <-> B =/= C ) )
33	29, 32	mpbird 240	. 2 |- ( ph -> ( F ` B ) =/= ( F ` C ) )
34	20, 28, 33	3jca 1210	1 |- ( ph -> ( ( F ` A ) =/= ( F ` B ) /\ ( F ` A ) =/= ( F ` C ) /\ ( F ` B ) =/= ( F ` C ) ) )

Syntax hints:   -> wi 4   <-> wb 189   \/ w3o 1006   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   {ctp 3963   -1-1->wf1 5586   ` cfv 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fv 5597

This theorem is referenced by:  f1dom3el3dif  6187