Theorem fodomb 4780

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 3663	. . . . . . . . . . . 12 |- (f:A-onto->B -> f:A-->B)
2		fdm 3623	. . . . . . . . . . . 12 |- (f:A-->B -> dom f = A)
3	1, 2	syl 10	. . . . . . . . . . 11 |- (f:A-onto->B -> dom f = A)
4	3	eqeq1d 1480	. . . . . . . . . 10 |- (f:A-onto->B -> (dom f = (/) <-> A = (/)))
5		forn 3665	. . . . . . . . . . . 12 |- (f:A-onto->B -> ran f = B)
6	5	eqeq1d 1480	. . . . . . . . . . 11 |- (f:A-onto->B -> (ran f = (/) <-> B = (/)))
7		dm0rn0 3325	. . . . . . . . . . 11 |- (dom f = (/) <-> ran f = (/))
8	6, 7	syl5bb 531	. . . . . . . . . 10 |- (f:A-onto->B -> (dom f = (/) <-> B = (/)))
9	4, 8	bitr3d 529	. . . . . . . . 9 |- (f:A-onto->B -> (A = (/) <-> B = (/)))
10	9	necon3bid 1598	. . . . . . . 8 |- (f:A-onto->B -> (A =/= (/) <-> B =/= (/)))
11	10	biimpac 418	. . . . . . 7 |- ((A =/= (/) /\ f:A-onto->B) -> B =/= (/))
12		fodomb.1	. . . . . . . . . 10 |- A e. V
13		fornex 3670	. . . . . . . . . 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 4452	. . . . . . . . 9 |- (B e. V -> ((/) ~< B <-> B =/= (/)))
16	14, 15	syl 10	. . . . . . . 8 |- (f:A-onto->B -> ((/) ~< B <-> B =/= (/)))
17	16	adantl 388	. . . . . . 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 4778	. . . . . 6 |- (f:A-onto->B -> B ~<_ A)
21	20	a1i 8	. . . . 5 |- (A =/= (/) -> (f:A-onto->B -> B ~<_ A))
22	19, 21	jcad 599	. . . 4 |- (A =/= (/) -> (f:A-onto->B -> ((/) ~< B /\ B ~<_ A)))
23	22	19.23adv 1212	. . 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 4455	. . . . 5 |- (A e. V -> (((/) ~< B /\ B ~<_ A) -> (/) ~< A))
26	12, 25	ax-mp 7	. . . 4 |- (((/) ~< B /\ B ~<_ A) -> (/) ~< A)
27	12	0sdom 4453	. . . 4 |- ((/) ~< A <-> A =/= (/))
28	26, 27	sylib 198	. . 3 |- (((/) ~< B /\ B ~<_ A) -> A =/= (/))
29		fodomr 4469	. . . 4 |- ((A e. V /\ (/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B)
30	12, 29	mp3an1 901	. . 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 954   e. wcel 956  E.wex 978   =/= wne 1582  Vcvv 1807  (/)c0 2276   class class class wbr 2614  dom cdm 3165  ran crn 3166  -->wf 3173  -onto->wfo 3175   ~<_ cdom 4355   ~< csdm 4356

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-er 4251  df-en 4357  df-dom 4358  df-sdom 4359

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