Theorem fodomfib 8240

Description: Equivalence of an onto mapping and dominance for a nonempty finite set. Unlike fodomb 9348 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof. (Contributed by NM, 23-Mar-2006.)

Assertion

Ref	Expression
fodomfib	|- ( A e. Fin -> ( ( A =/= (/) /\ E. f f : A -onto-> B ) <-> ( (/) ~< B /\ B ~<_ A ) ) )

Distinct variable groups:   A, f   B, f

Proof of Theorem fodomfib

Step	Hyp	Ref	Expression
1		fof 6115	. . . . . . . . . . . . 13 |- ( f : A -onto-> B -> f : A --> B )
2		fdm 6051	. . . . . . . . . . . . 13 |- ( f : A --> B -> dom f = A )
3	1, 2	syl 17	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> dom f = A )
4	3	eqeq1d 2624	. . . . . . . . . . 11 |- ( f : A -onto-> B -> ( dom f = (/) <-> A = (/) ) )
5		dm0rn0 5342	. . . . . . . . . . . 12 |- ( dom f = (/) <-> ran f = (/) )
6		forn 6118	. . . . . . . . . . . . 13 |- ( f : A -onto-> B -> ran f = B )
7	6	eqeq1d 2624	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> ( ran f = (/) <-> B = (/) ) )
8	5, 7	syl5bb 272	. . . . . . . . . . 11 |- ( f : A -onto-> B -> ( dom f = (/) <-> B = (/) ) )
9	4, 8	bitr3d 270	. . . . . . . . . 10 |- ( f : A -onto-> B -> ( A = (/) <-> B = (/) ) )
10	9	necon3bid 2838	. . . . . . . . 9 |- ( f : A -onto-> B -> ( A =/= (/) <-> B =/= (/) ) )
11	10	biimpac 503	. . . . . . . 8 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> B =/= (/) )
12	11	adantll 750	. . . . . . 7 |- ( ( ( A e. Fin /\ A =/= (/) ) /\ f : A -onto-> B ) -> B =/= (/) )
13		vex 3203	. . . . . . . . . . . 12 |- f e. _V
14	13	rnex 7100	. . . . . . . . . . 11 |- ran f e. _V
15	6, 14	syl6eqelr 2710	. . . . . . . . . 10 |- ( f : A -onto-> B -> B e. _V )
16	15	adantl 482	. . . . . . . . 9 |- ( ( A e. Fin /\ f : A -onto-> B ) -> B e. _V )
17		0sdomg 8089	. . . . . . . . 9 |- ( B e. _V -> ( (/) ~< B <-> B =/= (/) ) )
18	16, 17	syl 17	. . . . . . . 8 |- ( ( A e. Fin /\ f : A -onto-> B ) -> ( (/) ~< B <-> B =/= (/) ) )
19	18	adantlr 751	. . . . . . 7 |- ( ( ( A e. Fin /\ A =/= (/) ) /\ f : A -onto-> B ) -> ( (/) ~< B <-> B =/= (/) ) )
20	12, 19	mpbird 247	. . . . . 6 |- ( ( ( A e. Fin /\ A =/= (/) ) /\ f : A -onto-> B ) -> (/) ~< B )
21	20	ex 450	. . . . 5 |- ( ( A e. Fin /\ A =/= (/) ) -> ( f : A -onto-> B -> (/) ~< B ) )
22		fodomfi 8239	. . . . . . 7 |- ( ( A e. Fin /\ f : A -onto-> B ) -> B ~<_ A )
23	22	ex 450	. . . . . 6 |- ( A e. Fin -> ( f : A -onto-> B -> B ~<_ A ) )
24	23	adantr 481	. . . . 5 |- ( ( A e. Fin /\ A =/= (/) ) -> ( f : A -onto-> B -> B ~<_ A ) )
25	21, 24	jcad 555	. . . 4 |- ( ( A e. Fin /\ A =/= (/) ) -> ( f : A -onto-> B -> ( (/) ~< B /\ B ~<_ A ) ) )
26	25	exlimdv 1861	. . 3 |- ( ( A e. Fin /\ A =/= (/) ) -> ( E. f f : A -onto-> B -> ( (/) ~< B /\ B ~<_ A ) ) )
27	26	expimpd 629	. 2 |- ( A e. Fin -> ( ( A =/= (/) /\ E. f f : A -onto-> B ) -> ( (/) ~< B /\ B ~<_ A ) ) )
28		sdomdomtr 8093	. . . 4 |- ( ( (/) ~< B /\ B ~<_ A ) -> (/) ~< A )
29		0sdomg 8089	. . . 4 |- ( A e. Fin -> ( (/) ~< A <-> A =/= (/) ) )
30	28, 29	syl5ib 234	. . 3 |- ( A e. Fin -> ( ( (/) ~< B /\ B ~<_ A ) -> A =/= (/) ) )
31		fodomr 8111	. . . 4 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B )
32	31	a1i 11	. . 3 |- ( A e. Fin -> ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B ) )
33	30, 32	jcad 555	. 2 |- ( A e. Fin -> ( ( (/) ~< B /\ B ~<_ A ) -> ( A =/= (/) /\ E. f f : A -onto-> B ) ) )
34	27, 33	impbid 202	1 |- ( A e. Fin -> ( ( A =/= (/) /\ E. f f : A -onto-> B ) <-> ( (/) ~< B /\ B ~<_ A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ran crn 5115   -->wf 5884   -onto->wfo 5886   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955

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-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-rab 2921  df-v 3202  df-sbc 3436  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-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-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-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-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by: (None)