Theorem fodomb 5962

Description: Equivalence of an onto mapping and dominance for a non-empty set. Proposition 10.35 of [TakeutiZaring] p. 93.

Hypothesis

Ref	Expression
fodomb.1	|- A e. _V

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 4617	. . . . . . . . . . . 12 |- (f:A-onto->B -> f:A-->B)
2		fdm 4567	. . . . . . . . . . . 12 |- (f:A-->B -> dom f = A)
3	1, 2	syl 12	. . . . . . . . . . 11 |- (f:A-onto->B -> dom f = A)
4	3	eqeq1d 1892	. . . . . . . . . 10 |- (f:A-onto->B -> (dom f = (/) <-> A = (/)))
5		forn 4620	. . . . . . . . . . . 12 |- (f:A-onto->B -> ran f = B)
6	5	eqeq1d 1892	. . . . . . . . . . 11 |- (f:A-onto->B -> (ran f = (/) <-> B = (/)))
7		dm0rn0 4175	. . . . . . . . . . 11 |- (dom f = (/) <-> ran f = (/))
8	6, 7	syl5bb 591	. . . . . . . . . 10 |- (f:A-onto->B -> (dom f = (/) <-> B = (/)))
9	4, 8	bitr3d 589	. . . . . . . . 9 |- (f:A-onto->B -> (A = (/) <-> B = (/)))
10	9	necon3bid 2035	. . . . . . . 8 |- (f:A-onto->B -> (A =/= (/) <-> B =/= (/)))
11	10	biimpac 462	. . . . . . 7 |- ((A =/= (/) /\ f:A-onto->B) -> B =/= (/))
12		fodomb.1	. . . . . . . . . 10 |- A e. _V
13		fornex 4625	. . . . . . . . . 10 |- (A e. _V -> (f:A-onto->B -> B e. _V))
14	12, 13	ax-mp 7	. . . . . . . . 9 |- (f:A-onto->B -> B e. _V)
15		0sdomg 5529	. . . . . . . . 9 |- (B e. _V -> ((/) ~< B <-> B =/= (/)))
16	14, 15	syl 12	. . . . . . . 8 |- (f:A-onto->B -> ((/) ~< B <-> B =/= (/)))
17	16	adantl 424	. . . . . . 7 |- ((A =/= (/) /\ f:A-onto->B) -> ((/) ~< B <-> B =/= (/)))
18	11, 17	mpbird 213	. . . . . 6 |- ((A =/= (/) /\ f:A-onto->B) -> (/) ~< B)
19	18	ex 402	. . . . 5 |- (A =/= (/) -> (f:A-onto->B -> (/) ~< B))
20	12	fodom 5960	. . . . . 6 |- (f:A-onto->B -> B ~<_ A)
21	20	a1i 8	. . . . 5 |- (A =/= (/) -> (f:A-onto->B -> B ~<_ A))
22	19, 21	jcad 661	. . . 4 |- (A =/= (/) -> (f:A-onto->B -> ((/) ~< B /\ B ~<_ A)))
23	22	19.23adv 1584	. . 3 |- (A =/= (/) -> (E.f f:A-onto->B -> ((/) ~< B /\ B ~<_ A)))
24	23	imp 377	. 2 |- ((A =/= (/) /\ E.f f:A-onto->B) -> ((/) ~< B /\ B ~<_ A))
25		sdomdomtr 5532	. . . . 5 |- (A e. _V -> (((/) ~< B /\ B ~<_ A) -> (/) ~< A))
26	12, 25	ax-mp 7	. . . 4 |- (((/) ~< B /\ B ~<_ A) -> (/) ~< A)
27	12	0sdom 5530	. . . 4 |- ((/) ~< A <-> A =/= (/))
28	26, 27	sylib 215	. . 3 |- (((/) ~< B /\ B ~<_ A) -> A =/= (/))
29		fodomr 5547	. . . 4 |- ((A e. _V /\ (/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B)
30	12, 29	mp3an1 1178	. . 3 |- (((/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B)
31	28, 30	jca 310	. 2 |- (((/) ~< B /\ B ~<_ A) -> (A =/= (/) /\ E.f f:A-onto->B))
32	24, 31	impbii 174	1 |- ((A =/= (/) /\ E.f f:A-onto->B) <-> ((/) ~< B /\ B ~<_ A))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  _Vcvv 2292  (/)c0 2875   class class class wbr 3338  dom cdm 3986  ran crn 3987  -->wf 3994  -onto->wfo 3996   ~<_ cdom 5424   ~< csdm 5425

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429

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