Theorem fodomfi2 8883

Description: Onto functions define dominance when a finite number of choices need to be made. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Assertion

Ref	Expression
fodomfi2	|- ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) -> B ~<_ A )

Proof of Theorem fodomfi2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fofn 6117	. . . 4 |- ( F : A -onto-> B -> F Fn A )
2	1	3ad2ant3 1084	. . 3 |- ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) -> F Fn A )
3		forn 6118	. . . . 5 |- ( F : A -onto-> B -> ran F = B )
4		eqimss2 3658	. . . . 5 |- ( ran F = B -> B C_ ran F )
5	3, 4	syl 17	. . . 4 |- ( F : A -onto-> B -> B C_ ran F )
6	5	3ad2ant3 1084	. . 3 |- ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) -> B C_ ran F )
7		simp2 1062	. . 3 |- ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) -> B e. Fin )
8		fipreima 8272	. . 3 |- ( ( F Fn A /\ B C_ ran F /\ B e. Fin ) -> E. x e. ( ~P A i^i Fin ) ( F " x ) = B )
9	2, 6, 7, 8	syl3anc 1326	. 2 |- ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) -> E. x e. ( ~P A i^i Fin ) ( F " x ) = B )
10		inss2 3834	. . . . . . . . 9 |- ( ~P A i^i Fin ) C_ Fin
11	10	sseli 3599	. . . . . . . 8 |- ( x e. ( ~P A i^i Fin ) -> x e. Fin )
12	11	adantl 482	. . . . . . 7 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> x e. Fin )
13		finnum 8774	. . . . . . 7 |- ( x e. Fin -> x e. dom card )
14	12, 13	syl 17	. . . . . 6 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> x e. dom card )
15		simpl3 1066	. . . . . . . 8 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> F : A -onto-> B )
16		fofun 6116	. . . . . . . 8 |- ( F : A -onto-> B -> Fun F )
17	15, 16	syl 17	. . . . . . 7 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> Fun F )
18		inss1 3833	. . . . . . . . . . 11 |- ( ~P A i^i Fin ) C_ ~P A
19	18	sseli 3599	. . . . . . . . . 10 |- ( x e. ( ~P A i^i Fin ) -> x e. ~P A )
20	19	elpwid 4170	. . . . . . . . 9 |- ( x e. ( ~P A i^i Fin ) -> x C_ A )
21	20	adantl 482	. . . . . . . 8 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> x C_ A )
22		fof 6115	. . . . . . . . 9 |- ( F : A -onto-> B -> F : A --> B )
23		fdm 6051	. . . . . . . . 9 |- ( F : A --> B -> dom F = A )
24	15, 22, 23	3syl 18	. . . . . . . 8 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> dom F = A )
25	21, 24	sseqtr4d 3642	. . . . . . 7 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> x C_ dom F )
26		fores 6124	. . . . . . 7 |- ( ( Fun F /\ x C_ dom F ) -> ( F |` x ) : x -onto-> ( F " x ) )
27	17, 25, 26	syl2anc 693	. . . . . 6 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> ( F |` x ) : x -onto-> ( F " x ) )
28		fodomnum 8880	. . . . . 6 |- ( x e. dom card -> ( ( F |` x ) : x -onto-> ( F " x ) -> ( F " x ) ~<_ x ) )
29	14, 27, 28	sylc 65	. . . . 5 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> ( F " x ) ~<_ x )
30		simpl1 1064	. . . . . 6 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> A e. V )
31		ssdomg 8001	. . . . . 6 |- ( A e. V -> ( x C_ A -> x ~<_ A ) )
32	30, 21, 31	sylc 65	. . . . 5 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> x ~<_ A )
33		domtr 8009	. . . . 5 |- ( ( ( F " x ) ~<_ x /\ x ~<_ A ) -> ( F " x ) ~<_ A )
34	29, 32, 33	syl2anc 693	. . . 4 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> ( F " x ) ~<_ A )
35		breq1 4656	. . . 4 |- ( ( F " x ) = B -> ( ( F " x ) ~<_ A <-> B ~<_ A ) )
36	34, 35	syl5ibcom 235	. . 3 |- ( ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) /\ x e. ( ~P A i^i Fin ) ) -> ( ( F " x ) = B -> B ~<_ A ) )
37	36	rexlimdva 3031	. 2 |- ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) -> ( E. x e. ( ~P A i^i Fin ) ( F " x ) = B -> B ~<_ A ) )
38	9, 37	mpd 15	1 |- ( ( A e. V /\ B e. Fin /\ F : A -onto-> B ) -> B ~<_ A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ~<_ cdom 7953   Fincfn 7955   cardccrd 8761

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-1o 7560  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-fin 7959  df-card 8765  df-acn 8768

This theorem is referenced by:  wdomfil  8884