Theorem fodomb 8895

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 5777	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> f : A --> B )
2		fdm 5717	. . . . . . . . . . . 12 |- ( f : A --> B -> dom f = A )
3	1, 2	syl 16	. . . . . . . . . . 11 |- ( f : A -onto-> B -> dom f = A )
4	3	eqeq1d 2456	. . . . . . . . . 10 |- ( f : A -onto-> B -> ( dom f = (/) <-> A = (/) ) )
5		dm0rn0 5208	. . . . . . . . . . 11 |- ( dom f = (/) <-> ran f = (/) )
6		forn 5780	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> ran f = B )
7	6	eqeq1d 2456	. . . . . . . . . . 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 2712	. . . . . . . 8 |- ( f : A -onto-> B -> ( A =/= (/) <-> B =/= (/) ) )
11	10	biimpac 484	. . . . . . 7 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> B =/= (/) )
12		vex 3109	. . . . . . . . . . . 12 |- f e. _V
13	12	dmex 6706	. . . . . . . . . . 11 |- dom f e. _V
14	3, 13	syl6eqelr 2551	. . . . . . . . . 10 |- ( f : A -onto-> B -> A e. _V )
15		fornex 6742	. . . . . . . . . 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 7639	. . . . . . . . 9 |- ( B e. _V -> ( (/) ~< B <-> B =/= (/) ) )
18	16, 17	syl 16	. . . . . . . 8 |- ( f : A -onto-> B -> ( (/) ~< B <-> B =/= (/) ) )
19	18	adantl 464	. . . . . . 7 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> ( (/) ~< B <-> B =/= (/) ) )
20	11, 19	mpbird 232	. . . . . 6 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> (/) ~< B )
21	20	ex 432	. . . . 5 |- ( A =/= (/) -> ( f : A -onto-> B -> (/) ~< B ) )
22		fodomg 8894	. . . . . . 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 531	. . . 4 |- ( A =/= (/) -> ( f : A -onto-> B -> ( (/) ~< B /\ B ~<_ A ) ) )
26	25	exlimdv 1729	. . 3 |- ( A =/= (/) -> ( E. f f : A -onto-> B -> ( (/) ~< B /\ B ~<_ A ) ) )
27	26	imp 427	. 2 |- ( ( A =/= (/) /\ E. f f : A -onto-> B ) -> ( (/) ~< B /\ B ~<_ A ) )
28		sdomdomtr 7643	. . . 4 |- ( ( (/) ~< B /\ B ~<_ A ) -> (/) ~< A )
29		reldom 7515	. . . . . . 7 |- Rel ~<_
30	29	brrelex2i 5030	. . . . . 6 |- ( B ~<_ A -> A e. _V )
31	30	adantl 464	. . . . 5 |- ( ( (/) ~< B /\ B ~<_ A ) -> A e. _V )
32		0sdomg 7639	. . . . 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 7661	. . 3 |- ( ( (/) ~< B /\ B ~<_ A ) -> E. f f : A -onto-> B )
36	34, 35	jca 530	. 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 367   = wceq 1398   E.wex 1617   e. wcel 1823   =/= wne 2649   _Vcvv 3106   (/)c0 3783   class class class wbr 4439   dom cdm 4988   ran crn 4989   -->wf 5566   -onto->wfo 5568   ~<_ cdom 7507   ~< csdm 7508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-ac2 8834

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-recs 7034  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-card 8311  df-acn 8314  df-ac 8488

This theorem is referenced by: (None)