Theorem fodomb 8797

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 5721	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> f : A --> B )
2		fdm 5664	. . . . . . . . . . . 12 |- ( f : A --> B -> dom f = A )
3	1, 2	syl 16	. . . . . . . . . . 11 |- ( f : A -onto-> B -> dom f = A )
4	3	eqeq1d 2453	. . . . . . . . . 10 |- ( f : A -onto-> B -> ( dom f = (/) <-> A = (/) ) )
5		dm0rn0 5157	. . . . . . . . . . 11 |- ( dom f = (/) <-> ran f = (/) )
6		forn 5724	. . . . . . . . . . . 12 |- ( f : A -onto-> B -> ran f = B )
7	6	eqeq1d 2453	. . . . . . . . . . 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 2706	. . . . . . . 8 |- ( f : A -onto-> B -> ( A =/= (/) <-> B =/= (/) ) )
11	10	biimpac 486	. . . . . . 7 |- ( ( A =/= (/) /\ f : A -onto-> B ) -> B =/= (/) )
12		vex 3074	. . . . . . . . . . . 12 |- f e. _V
13	12	dmex 6614	. . . . . . . . . . 11 |- dom f e. _V
14	3, 13	syl6eqelr 2548	. . . . . . . . . 10 |- ( f : A -onto-> B -> A e. _V )
15		fornex 6649	. . . . . . . . . 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 7543	. . . . . . . . 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 8796	. . . . . . 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 1691	. . 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 7547	. . . 4 |- ( ( (/) ~< B /\ B ~<_ A ) -> (/) ~< A )
29		reldom 7419	. . . . . . 7 |- Rel ~<_
30	29	brrelex2i 4981	. . . . . 6 |- ( B ~<_ A -> A e. _V )
31	30	adantl 466	. . . . 5 |- ( ( (/) ~< B /\ B ~<_ A ) -> A e. _V )
32		0sdomg 7543	. . . . 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 7565	. . 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 1370   E.wex 1587   e. wcel 1758   =/= wne 2644   _Vcvv 3071   (/)c0 3738   class class class wbr 4393   dom cdm 4941   ran crn 4942   -->wf 5515   -onto->wfo 5517   ~<_ cdom 7411   ~< csdm 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-ac2 8736

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-recs 6935  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-card 8213  df-acn 8216  df-ac 8390

This theorem is referenced by: (None)