Theorem fodomfib 5657

Description: Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 5962 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof.

Assertion

Ref	Expression
fodomfib	|- (A e. Fin -> ((A =/= (/) /\ E.f f:A-onto->B) <-> ((/) ~< B /\ B ~<_ A)))

Distinct variable groups:   A,f   B,f

Proof of Theorem fodomfib

Step	Hyp	Ref	Expression
1		fof 4617	. . . . . . . . . . . . 13 |- (f:A-onto->B -> f:A-->B)
2		fdm 4567	. . . . . . . . . . . . 13 |- (f:A-->B -> dom f = A)
3	1, 2	syl 12	. . . . . . . . . . . 12 |- (f:A-onto->B -> dom f = A)
4	3	eqeq1d 1892	. . . . . . . . . . 11 |- (f:A-onto->B -> (dom f = (/) <-> A = (/)))
5		forn 4620	. . . . . . . . . . . . 13 |- (f:A-onto->B -> ran f = B)
6	5	eqeq1d 1892	. . . . . . . . . . . 12 |- (f:A-onto->B -> (ran f = (/) <-> B = (/)))
7		dm0rn0 4175	. . . . . . . . . . . 12 |- (dom f = (/) <-> ran f = (/))
8	6, 7	syl5bb 591	. . . . . . . . . . 11 |- (f:A-onto->B -> (dom f = (/) <-> B = (/)))
9	4, 8	bitr3d 589	. . . . . . . . . 10 |- (f:A-onto->B -> (A = (/) <-> B = (/)))
10	9	necon3bid 2035	. . . . . . . . 9 |- (f:A-onto->B -> (A =/= (/) <-> B =/= (/)))
11	10	biimpac 462	. . . . . . . 8 |- ((A =/= (/) /\ f:A-onto->B) -> B =/= (/))
12	11	adantll 428	. . . . . . 7 |- (((A e. Fin /\ A =/= (/)) /\ f:A-onto->B) -> B =/= (/))
13		fornex 4625	. . . . . . . . . 10 |- (A e. Fin -> (f:A-onto->B -> B e. _V))
14	13	imp 377	. . . . . . . . 9 |- ((A e. Fin /\ f:A-onto->B) -> B e. _V)
15		0sdomg 5529	. . . . . . . . 9 |- (B e. _V -> ((/) ~< B <-> B =/= (/)))
16	14, 15	syl 12	. . . . . . . 8 |- ((A e. Fin /\ f:A-onto->B) -> ((/) ~< B <-> B =/= (/)))
17	16	adantlr 429	. . . . . . 7 |- (((A e. Fin /\ A =/= (/)) /\ f:A-onto->B) -> ((/) ~< B <-> B =/= (/)))
18	12, 17	mpbird 213	. . . . . 6 |- (((A e. Fin /\ A =/= (/)) /\ f:A-onto->B) -> (/) ~< B)
19	18	ex 402	. . . . 5 |- ((A e. Fin /\ A =/= (/)) -> (f:A-onto->B -> (/) ~< B))
20		fodomfi 5656	. . . . . . 7 |- ((A e. Fin /\ f:A-onto->B) -> B ~<_ A)
21	20	ex 402	. . . . . 6 |- (A e. Fin -> (f:A-onto->B -> B ~<_ A))
22	21	adantr 425	. . . . 5 |- ((A e. Fin /\ A =/= (/)) -> (f:A-onto->B -> B ~<_ A))
23	19, 22	jcad 661	. . . 4 |- ((A e. Fin /\ A =/= (/)) -> (f:A-onto->B -> ((/) ~< B /\ B ~<_ A)))
24	23	19.23adv 1584	. . 3 |- ((A e. Fin /\ A =/= (/)) -> (E.f f:A-onto->B -> ((/) ~< B /\ B ~<_ A)))
25	24	expimpd 404	. 2 |- (A e. Fin -> ((A =/= (/) /\ E.f f:A-onto->B) -> ((/) ~< B /\ B ~<_ A)))
26		sdomdomtr 5532	. . . 4 |- (A e. Fin -> (((/) ~< B /\ B ~<_ A) -> (/) ~< A))
27		0sdomg 5529	. . . 4 |- (A e. Fin -> ((/) ~< A <-> A =/= (/)))
28	26, 27	sylibd 219	. . 3 |- (A e. Fin -> (((/) ~< B /\ B ~<_ A) -> A =/= (/)))
29		fodomr 5547	. . . 4 |- ((A e. Fin /\ (/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B)
30	29	3expib 1070	. . 3 |- (A e. Fin -> (((/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B))
31	28, 30	jcad 661	. 2 |- (A e. Fin -> (((/) ~< B /\ B ~<_ A) -> (A =/= (/) /\ E.f f:A-onto->B)))
32	25, 31	impbid 574	1 |- (A e. Fin -> ((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  Fincfn 5426

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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  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-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-1o 5177  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430

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