Theorem fodomb 8907

Description: Equivalence of an onto mapping and dominance for a nonempty set. Proposition 10.35 of [TakeutiZaring] p. 93. (Contributed by NM, 29-Jul-2004.)

Assertion

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

Distinct variable groups:   A, f   B, f

Proof of Theorem fodomb

Step	Hyp	Ref	Expression
1		fof 5785	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> f : A --> B )
2		fdm 5725	. . . . . . . . . . . 12 |- ( f : A --> B -> dom f = A )
3	1, 2	syl 16	. . . . . . . . . . 11 |- ( f : A -onto-> B -> dom f = A )
4	3	eqeq1d 2445	. . . . . . . . . 10 |- ( f : A -onto-> B -> ( dom f = (/) <-> A = (/) ) )
5		dm0rn0 5209	. . . . . . . . . . 11 |- ( dom f = (/) <-> ran f = (/) )
6		forn 5788	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> ran f = B )
7	6	eqeq1d 2445	. . . . . . . . . . 11 |- ( f : A -onto-> B -> ( ran f = (/) <-> B = (/) ) )
8	5, 7	syl5bb 257	. . . . . . . . . 10 |- ( f : A -onto-> B -> ( dom f = (/) <-> B = (/) ) )
9	4, 8	bitr3d 255	. . . . . . . . 9 |- ( f : A -onto-> B -> ( A = (/) <-> B = (/) ) )
10	9	necon3bid 2701	. . . . . . . 8 |- ( f : A -onto-> B -> ( A =/= (/) <-> B =/= (/) ) )
11	10	biimpac 486	. . . . . . 7 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> B =/= (/) )
12		vex 3098	. . . . . . . . . . . 12 |- f e. _V
13	12	dmex 6718	. . . . . . . . . . 11 |- dom f e. _V
14	3, 13	syl6eqelr 2540	. . . . . . . . . 10 |- ( f : A -onto-> B -> A e. _V )
15		fornex 6754	. . . . . . . . . 10 |- ( A e. _V -> ( f : A -onto-> B -> B e. _V ) )
16	14, 15	mpcom 36	. . . . . . . . 9 |- ( f : A -onto-> B -> B e. _V )
17		0sdomg 7648	. . . . . . . . 9 |- ( B e. _V -> ( (/) ~< B <-> B =/= (/) ) )
18	16, 17	syl 16	. . . . . . . 8 |- ( f : A -onto-> B -> ( (/) ~< B <-> B =/= (/) ) )
19	18	adantl 466	. . . . . . 7 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> ( (/) ~< B <-> B =/= (/) ) )
20	11, 19	mpbird 232	. . . . . 6 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> (/) ~< B )
21	20	ex 434	. . . . 5 |- ( A =/= (/) -> ( f : A -onto-> B -> (/) ~< B ) )
22		fodomg 8906	. . . . . . 7 |- ( A e. _V -> ( f : A -onto-> B -> B ~<_ A ) )
23	14, 22	mpcom 36	. . . . . 6 |- ( f : A -onto-> B -> B ~<_ A )
24	23	a1i 11	. . . . 5 |- ( A =/= (/) -> ( f : A -onto-> B -> B ~<_ A ) )
25	21, 24	jcad 533	. . . 4 |- ( A =/= (/) -> ( f : A -onto-> B -> ( (/) ~< B /\ B ~<_ A ) ) )
26	25	exlimdv 1711	. . 3 |- ( A =/= (/) -> ( E. f f : A -onto-> B -> ( (/) ~< B /\ B ~<_ A ) ) )
27	26	imp 429	. 2 |- ( ( A =/= (/) /\ E. f f : A -onto-> B ) -> ( (/) ~< B /\ B ~<_ A ) )
28		sdomdomtr 7652	. . . 4 |- ( ( (/) ~< B /\ B ~<_ A ) -> (/) ~< A )
29		reldom 7524	. . . . . . 7 |- Rel ~<_
30	29	brrelex2i 5031	. . . . . 6 |- ( B ~<_ A -> A e. _V )
31	30	adantl 466	. . . . 5 |- ( ( (/) ~< B /\ B ~<_ A ) -> A e. _V )
32		0sdomg 7648	. . . . 5 |- ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) )
33	31, 32	syl 16	. . . 4 |- ( ( (/) ~< B /\ B ~<_ A ) -> ( (/) ~< A <-> A =/= (/) ) )
34	28, 33	mpbid 210	. . 3 |- ( ( (/) ~< B /\ B ~<_ A ) -> A =/= (/) )
35		fodomr 7670	. . 3 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B )
36	34, 35	jca 532	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> ( A =/= (/) /\ E. f f : A -onto-> B ) )
37	27, 36	impbii 188	1 |- ( ( A =/= (/) /\ E. f f : A -onto-> B ) <-> ( (/) ~< B /\ B ~<_ A ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1383   E.wex 1599   e. wcel 1804   =/= wne 2638   _Vcvv 3095   (/)c0 3770   class class class wbr 4437   dom cdm 4989   ran crn 4990   -->wf 5574   -onto->wfo 5576   ~<_ cdom 7516   ~< csdm 7517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-ac2 8846

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-recs 7044  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-card 8323  df-acn 8326  df-ac 8500

This theorem is referenced by: (None)