Theorem fodomb 4810

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 3678	. . . . . . . . . . . 12 |- (f:A-onto->B -> f:A-->B)
2		fdm 3637	. . . . . . . . . . . 12 |- (f:A-->B -> dom f = A)
3	1, 2	syl 10	. . . . . . . . . . 11 |- (f:A-onto->B -> dom f = A)
4	3	eqeq1d 1486	. . . . . . . . . 10 |- (f:A-onto->B -> (dom f = (/) <-> A = (/)))
5		forn 3680	. . . . . . . . . . . 12 |- (f:A-onto->B -> ran f = B)
6	5	eqeq1d 1486	. . . . . . . . . . 11 |- (f:A-onto->B -> (ran f = (/) <-> B = (/)))
7		dm0rn0 3336	. . . . . . . . . . 11 |- (dom f = (/) <-> ran f = (/))
8	6, 7	syl5bb 534	. . . . . . . . . 10 |- (f:A-onto->B -> (dom f = (/) <-> B = (/)))
9	4, 8	bitr3d 532	. . . . . . . . 9 |- (f:A-onto->B -> (A = (/) <-> B = (/)))
10	9	necon3bid 1604	. . . . . . . 8 |- (f:A-onto->B -> (A =/= (/) <-> B =/= (/)))
11	10	biimpac 420	. . . . . . 7 |- ((A =/= (/) /\ f:A-onto->B) -> B =/= (/))
12		fodomb.1	. . . . . . . . . 10 |- A e. V
13		fornex 3685	. . . . . . . . . 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 4472	. . . . . . . . 9 |- (B e. V -> ((/) ~< B <-> B =/= (/)))
16	14, 15	syl 10	. . . . . . . 8 |- (f:A-onto->B -> ((/) ~< B <-> B =/= (/)))
17	16	adantl 390	. . . . . . 7 |- ((A =/= (/) /\ f:A-onto->B) -> ((/) ~< B <-> B =/= (/)))
18	11, 17	mpbird 196	. . . . . 6 |- ((A =/= (/) /\ f:A-onto->B) -> (/) ~< B)
19	18	ex 373	. . . . 5 |- (A =/= (/) -> (f:A-onto->B -> (/) ~< B))
20	12	fodom 4808	. . . . . 6 |- (f:A-onto->B -> B ~<_ A)
21	20	a1i 8	. . . . 5 |- (A =/= (/) -> (f:A-onto->B -> B ~<_ A))
22	19, 21	jcad 602	. . . 4 |- (A =/= (/) -> (f:A-onto->B -> ((/) ~< B /\ B ~<_ A)))
23	22	19.23adv 1216	. . 3 |- (A =/= (/) -> (E.f f:A-onto->B -> ((/) ~< B /\ B ~<_ A)))
24	23	imp 350	. 2 |- ((A =/= (/) /\ E.f f:A-onto->B) -> ((/) ~< B /\ B ~<_ A))
25		sdomdomtr 4475	. . . . 5 |- (A e. V -> (((/) ~< B /\ B ~<_ A) -> (/) ~< A))
26	12, 25	ax-mp 7	. . . 4 |- (((/) ~< B /\ B ~<_ A) -> (/) ~< A)
27	12	0sdom 4473	. . . 4 |- ((/) ~< A <-> A =/= (/))
28	26, 27	sylib 198	. . 3 |- (((/) ~< B /\ B ~<_ A) -> A =/= (/))
29		fodomr 4489	. . . 4 |- ((A e. V /\ (/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B)
30	12, 29	mp3an1 905	. . 3 |- (((/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B)
31	28, 30	jca 288	. 2 |- (((/) ~< B /\ B ~<_ A) -> (A =/= (/) /\ E.f f:A-onto->B))
32	24, 31	impbi 157	1 |- ((A =/= (/) /\ E.f f:A-onto->B) <-> ((/) ~< B /\ B ~<_ A))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982   =/= wne 1588  Vcvv 1814  (/)c0 2283   class class class wbr 2624  dom cdm 3176  ran crn 3177  -->wf 3184  -onto->wfo 3186   ~<_ cdom 4371   ~< csdm 4372

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-er 4267  df-en 4374  df-dom 4375  df-sdom 4376

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