Theorem fodomb 8954

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 5793	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> f : A --> B )
2		fdm 5733	. . . . . . . . . . . 12 |- ( f : A --> B -> dom f = A )
3	1, 2	syl 17	. . . . . . . . . . 11 |- ( f : A -onto-> B -> dom f = A )
4	3	eqeq1d 2453	. . . . . . . . . 10 |- ( f : A -onto-> B -> ( dom f = (/) <-> A = (/) ) )
5		dm0rn0 5051	. . . . . . . . . . 11 |- ( dom f = (/) <-> ran f = (/) )
6		forn 5796	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> ran f = B )
7	6	eqeq1d 2453	. . . . . . . . . . 11 |- ( f : A -onto-> B -> ( ran f = (/) <-> B = (/) ) )
8	5, 7	syl5bb 261	. . . . . . . . . 10 |- ( f : A -onto-> B -> ( dom f = (/) <-> B = (/) ) )
9	4, 8	bitr3d 259	. . . . . . . . 9 |- ( f : A -onto-> B -> ( A = (/) <-> B = (/) ) )
10	9	necon3bid 2668	. . . . . . . 8 |- ( f : A -onto-> B -> ( A =/= (/) <-> B =/= (/) ) )
11	10	biimpac 489	. . . . . . 7 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> B =/= (/) )
12		vex 3048	. . . . . . . . . . . 12 |- f e. _V
13	12	dmex 6726	. . . . . . . . . . 11 |- dom f e. _V
14	3, 13	syl6eqelr 2538	. . . . . . . . . 10 |- ( f : A -onto-> B -> A e. _V )
15		fornex 6762	. . . . . . . . . 10 |- ( A e. _V -> ( f : A -onto-> B -> B e. _V ) )
16	14, 15	mpcom 37	. . . . . . . . 9 |- ( f : A -onto-> B -> B e. _V )
17		0sdomg 7701	. . . . . . . . 9 |- ( B e. _V -> ( (/) ~< B <-> B =/= (/) ) )
18	16, 17	syl 17	. . . . . . . 8 |- ( f : A -onto-> B -> ( (/) ~< B <-> B =/= (/) ) )
19	18	adantl 468	. . . . . . 7 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> ( (/) ~< B <-> B =/= (/) ) )
20	11, 19	mpbird 236	. . . . . 6 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> (/) ~< B )
21	20	ex 436	. . . . 5 |- ( A =/= (/) -> ( f : A -onto-> B -> (/) ~< B ) )
22		fodomg 8953	. . . . . . 7 |- ( A e. _V -> ( f : A -onto-> B -> B ~<_ A ) )
23	14, 22	mpcom 37	. . . . . 6 |- ( f : A -onto-> B -> B ~<_ A )
24	23	a1i 11	. . . . 5 |- ( A =/= (/) -> ( f : A -onto-> B -> B ~<_ A ) )
25	21, 24	jcad 536	. . . 4 |- ( A =/= (/) -> ( f : A -onto-> B -> ( (/) ~< B /\ B ~<_ A ) ) )
26	25	exlimdv 1779	. . 3 |- ( A =/= (/) -> ( E. f f : A -onto-> B -> ( (/) ~< B /\ B ~<_ A ) ) )
27	26	imp 431	. 2 |- ( ( A =/= (/) /\ E. f f : A -onto-> B ) -> ( (/) ~< B /\ B ~<_ A ) )
28		sdomdomtr 7705	. . . 4 |- ( ( (/) ~< B /\ B ~<_ A ) -> (/) ~< A )
29		reldom 7575	. . . . . . 7 |- Rel ~<_
30	29	brrelex2i 4876	. . . . . 6 |- ( B ~<_ A -> A e. _V )
31	30	adantl 468	. . . . 5 |- ( ( (/) ~< B /\ B ~<_ A ) -> A e. _V )
32		0sdomg 7701	. . . . 5 |- ( A e. _V -> ( (/) ~< A <-> A =/= (/) ) )
33	31, 32	syl 17	. . . 4 |- ( ( (/) ~< B /\ B ~<_ A ) -> ( (/) ~< A <-> A =/= (/) ) )
34	28, 33	mpbid 214	. . 3 |- ( ( (/) ~< B /\ B ~<_ A ) -> A =/= (/) )
35		fodomr 7723	. . 3 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B )
36	34, 35	jca 535	. 2 |- ( ( (/) ~< B /\ B ~<_ A ) -> ( A =/= (/) /\ E. f f : A -onto-> B ) )
37	27, 36	impbii 191	1 |- ( ( A =/= (/) /\ E. f f : A -onto-> B ) <-> ( (/) ~< B /\ B ~<_ A ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   E.wex 1663   e. wcel 1887   =/= wne 2622   _Vcvv 3045   (/)c0 3731   class class class wbr 4402   dom cdm 4834   ran crn 4835   -->wf 5578   -onto->wfo 5580   ~<_ cdom 7567   ~< csdm 7568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-ac2 8893

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-card 8373  df-acn 8376  df-ac 8547

This theorem is referenced by: (None)