Theorem fodomb 8893

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 5786	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> f : A --> B )
2		fdm 5726	. . . . . . . . . . . 12 |- ( f : A --> B -> dom f = A )
3	1, 2	syl 16	. . . . . . . . . . 11 |- ( f : A -onto-> B -> dom f = A )
4	3	eqeq1d 2462	. . . . . . . . . 10 |- ( f : A -onto-> B -> ( dom f = (/) <-> A = (/) ) )
5		dm0rn0 5210	. . . . . . . . . . 11 |- ( dom f = (/) <-> ran f = (/) )
6		forn 5789	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> ran f = B )
7	6	eqeq1d 2462	. . . . . . . . . . 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 2718	. . . . . . . 8 |- ( f : A -onto-> B -> ( A =/= (/) <-> B =/= (/) ) )
11	10	biimpac 486	. . . . . . 7 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> B =/= (/) )
12		vex 3109	. . . . . . . . . . . 12 |- f e. _V
13	12	dmex 6707	. . . . . . . . . . 11 |- dom f e. _V
14	3, 13	syl6eqelr 2557	. . . . . . . . . 10 |- ( f : A -onto-> B -> A e. _V )
15		fornex 6743	. . . . . . . . . 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 7636	. . . . . . . . 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 8892	. . . . . . 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 1695	. . 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 7640	. . . 4 |- ( ( (/) ~< B /\ B ~<_ A ) -> (/) ~< A )
29		reldom 7512	. . . . . . 7 |- Rel ~<_
30	29	brrelex2i 5033	. . . . . 6 |- ( B ~<_ A -> A e. _V )
31	30	adantl 466	. . . . 5 |- ( ( (/) ~< B /\ B ~<_ A ) -> A e. _V )
32		0sdomg 7636	. . . . 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 7658	. . 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 1374   E.wex 1591   e. wcel 1762   =/= wne 2655   _Vcvv 3106   (/)c0 3778   class class class wbr 4440   dom cdm 4992   ran crn 4993   -->wf 5575   -onto->wfo 5577   ~<_ cdom 7504   ~< csdm 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-ac2 8832

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-recs 7032  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-card 8309  df-acn 8312  df-ac 8486

This theorem is referenced by: (None)