Theorem f1dom3el3dif 6526

Description: The range of a 1-1 function from a set with three different elements has (at least) three different elements. (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
f1dom3el3dif	|- ( ph -> E. x e. R E. y e. R E. z e. R ( x =/= y /\ x =/= z /\ y =/= z ) )

Distinct variable groups:   x, A, y, z   x, B, y, z   x, C, y, z   x, F, y, z   x, R, y, z

Allowed substitution hints:   ph( x, y, z)   X( x, y, z)   Y( x, y, z)   Z( x, y, z)

Proof of Theorem f1dom3el3dif

Step	Hyp	Ref	Expression
1		f1dom3fv3dif.f	. . 3 |- ( ph -> F : { A , B , C } -1-1-> R )
2		f1f 6101	. . . 4 |- ( F : { A , B , C } -1-1-> R -> F : { A , B , C } --> R )
3		simpr 477	. . . . . . 7 |- ( ( ph /\ F : { A , B , C } --> R ) -> F : { A , B , C } --> R )
4		eqidd 2623	. . . . . . . . . 10 |- ( ph -> A = A )
5	4	3mix1d 1236	. . . . . . . . 9 |- ( ph -> ( A = A \/ A = B \/ A = C ) )
6		f1dom3fv3dif.v	. . . . . . . . . . 11 |- ( ph -> ( A e. X /\ B e. Y /\ C e. Z ) )
7	6	simp1d 1073	. . . . . . . . . 10 |- ( ph -> A e. X )
8		eltpg 4227	. . . . . . . . . 10 |- ( A e. X -> ( A e. { A , B , C } <-> ( A = A \/ A = B \/ A = C ) ) )
9	7, 8	syl 17	. . . . . . . . 9 |- ( ph -> ( A e. { A , B , C } <-> ( A = A \/ A = B \/ A = C ) ) )
10	5, 9	mpbird 247	. . . . . . . 8 |- ( ph -> A e. { A , B , C } )
11	10	adantr 481	. . . . . . 7 |- ( ( ph /\ F : { A , B , C } --> R ) -> A e. { A , B , C } )
12	3, 11	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ F : { A , B , C } --> R ) -> ( F ` A ) e. R )
13		eqidd 2623	. . . . . . . . . 10 |- ( ph -> B = B )
14	13	3mix2d 1237	. . . . . . . . 9 |- ( ph -> ( B = A \/ B = B \/ B = C ) )
15	6	simp2d 1074	. . . . . . . . . 10 |- ( ph -> B e. Y )
16		eltpg 4227	. . . . . . . . . 10 |- ( B e. Y -> ( B e. { A , B , C } <-> ( B = A \/ B = B \/ B = C ) ) )
17	15, 16	syl 17	. . . . . . . . 9 |- ( ph -> ( B e. { A , B , C } <-> ( B = A \/ B = B \/ B = C ) ) )
18	14, 17	mpbird 247	. . . . . . . 8 |- ( ph -> B e. { A , B , C } )
19	18	adantr 481	. . . . . . 7 |- ( ( ph /\ F : { A , B , C } --> R ) -> B e. { A , B , C } )
20	3, 19	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ F : { A , B , C } --> R ) -> ( F ` B ) e. R )
21	6	simp3d 1075	. . . . . . . . 9 |- ( ph -> C e. Z )
22		tpid3g 4305	. . . . . . . . 9 |- ( C e. Z -> C e. { A , B , C } )
23	21, 22	syl 17	. . . . . . . 8 |- ( ph -> C e. { A , B , C } )
24	23	adantr 481	. . . . . . 7 |- ( ( ph /\ F : { A , B , C } --> R ) -> C e. { A , B , C } )
25	3, 24	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ F : { A , B , C } --> R ) -> ( F ` C ) e. R )
26	12, 20, 25	3jca 1242	. . . . 5 |- ( ( ph /\ F : { A , B , C } --> R ) -> ( ( F ` A ) e. R /\ ( F ` B ) e. R /\ ( F ` C ) e. R ) )
27	26	expcom 451	. . . 4 |- ( F : { A , B , C } --> R -> ( ph -> ( ( F ` A ) e. R /\ ( F ` B ) e. R /\ ( F ` C ) e. R ) ) )
28	2, 27	syl 17	. . 3 |- ( F : { A , B , C } -1-1-> R -> ( ph -> ( ( F ` A ) e. R /\ ( F ` B ) e. R /\ ( F ` C ) e. R ) ) )
29	1, 28	mpcom 38	. 2 |- ( ph -> ( ( F ` A ) e. R /\ ( F ` B ) e. R /\ ( F ` C ) e. R ) )
30		f1dom3fv3dif.n	. . 3 |- ( ph -> ( A =/= B /\ A =/= C /\ B =/= C ) )
31	6, 30, 1	f1dom3fv3dif 6525	. 2 |- ( ph -> ( ( F ` A ) =/= ( F ` B ) /\ ( F ` A ) =/= ( F ` C ) /\ ( F ` B ) =/= ( F ` C ) ) )
32		neeq1 2856	. . . 4 |- ( x = ( F ` A ) -> ( x =/= y <-> ( F ` A ) =/= y ) )
33		neeq1 2856	. . . 4 |- ( x = ( F ` A ) -> ( x =/= z <-> ( F ` A ) =/= z ) )
34	32, 33	3anbi12d 1400	. . 3 |- ( x = ( F ` A ) -> ( ( x =/= y /\ x =/= z /\ y =/= z ) <-> ( ( F ` A ) =/= y /\ ( F ` A ) =/= z /\ y =/= z ) ) )
35		neeq2 2857	. . . 4 |- ( y = ( F ` B ) -> ( ( F ` A ) =/= y <-> ( F ` A ) =/= ( F ` B ) ) )
36		neeq1 2856	. . . 4 |- ( y = ( F ` B ) -> ( y =/= z <-> ( F ` B ) =/= z ) )
37	35, 36	3anbi13d 1401	. . 3 |- ( y = ( F ` B ) -> ( ( ( F ` A ) =/= y /\ ( F ` A ) =/= z /\ y =/= z ) <-> ( ( F ` A ) =/= ( F ` B ) /\ ( F ` A ) =/= z /\ ( F ` B ) =/= z ) ) )
38		neeq2 2857	. . . 4 |- ( z = ( F ` C ) -> ( ( F ` A ) =/= z <-> ( F ` A ) =/= ( F ` C ) ) )
39		neeq2 2857	. . . 4 |- ( z = ( F ` C ) -> ( ( F ` B ) =/= z <-> ( F ` B ) =/= ( F ` C ) ) )
40	38, 39	3anbi23d 1402	. . 3 |- ( z = ( F ` C ) -> ( ( ( F ` A ) =/= ( F ` B ) /\ ( F ` A ) =/= z /\ ( F ` B ) =/= z ) <-> ( ( F ` A ) =/= ( F ` B ) /\ ( F ` A ) =/= ( F ` C ) /\ ( F ` B ) =/= ( F ` C ) ) ) )
41	34, 37, 40	rspc3ev 3326	. 2 |- ( ( ( ( F ` A ) e. R /\ ( F ` B ) e. R /\ ( F ` C ) e. R ) /\ ( ( F ` A ) =/= ( F ` B ) /\ ( F ` A ) =/= ( F ` C ) /\ ( F ` B ) =/= ( F ` C ) ) ) -> E. x e. R E. y e. R E. z e. R ( x =/= y /\ x =/= z /\ y =/= z ) )
42	29, 31, 41	syl2anc 693	1 |- ( ph -> E. x e. R E. y e. R E. z e. R ( x =/= y /\ x =/= z /\ y =/= z ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {ctp 4181   -->wf 5884   -1-1->wf1 5885   ` 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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896

This theorem is referenced by:  hashge3el3dif  13268