Theorem fodomfib 4541

Description: Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 4772 for arbitrary sets, this theorem does not require the Axiom of Choice for its proof.

Assertion

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

Distinct variable groups:   f,n,A   B,f

Proof of Theorem fodomfib

Step	Hyp	Ref	Expression
1		fof 3657	. . . . . . . . . . . . . . 15 |- (f:A-onto->B -> f:A-->B)
2		fdm 3617	. . . . . . . . . . . . . . 15 |- (f:A-->B -> dom f = A)
3	1, 2	syl 10	. . . . . . . . . . . . . 14 |- (f:A-onto->B -> dom f = A)
4	3	eqeq1d 1475	. . . . . . . . . . . . 13 |- (f:A-onto->B -> (dom f = (/) <-> A = (/)))
5		forn 3659	. . . . . . . . . . . . . . 15 |- (f:A-onto->B -> ran f = B)
6	5	eqeq1d 1475	. . . . . . . . . . . . . 14 |- (f:A-onto->B -> (ran f = (/) <-> B = (/)))
7		dm0rn0 3319	. . . . . . . . . . . . . 14 |- (dom f = (/) <-> ran f = (/))
8	6, 7	syl5bb 530	. . . . . . . . . . . . 13 |- (f:A-onto->B -> (dom f = (/) <-> B = (/)))
9	4, 8	bitr3d 528	. . . . . . . . . . . 12 |- (f:A-onto->B -> (A = (/) <-> B = (/)))
10	9	necon3bid 1593	. . . . . . . . . . 11 |- (f:A-onto->B -> (A =/= (/) <-> B =/= (/)))
11	10	biimpac 418	. . . . . . . . . 10 |- ((A =/= (/) /\ f:A-onto->B) -> B =/= (/))
12	11	adantll 392	. . . . . . . . 9 |- (((A e. V /\ A =/= (/)) /\ f:A-onto->B) -> B =/= (/))
13		fornex 3664	. . . . . . . . . . . 12 |- (A e. V -> (f:A-onto->B -> B e. V))
14	13	imp 350	. . . . . . . . . . 11 |- ((A e. V /\ f:A-onto->B) -> B e. V)
15		0sdomg 4446	. . . . . . . . . . 11 |- (B e. V -> ((/) ~< B <-> B =/= (/)))
16	14, 15	syl 10	. . . . . . . . . 10 |- ((A e. V /\ f:A-onto->B) -> ((/) ~< B <-> B =/= (/)))
17	16	adantlr 393	. . . . . . . . 9 |- (((A e. V /\ A =/= (/)) /\ f:A-onto->B) -> ((/) ~< B <-> B =/= (/)))
18	12, 17	mpbird 196	. . . . . . . 8 |- (((A e. V /\ A =/= (/)) /\ f:A-onto->B) -> (/) ~< B)
19	18	ex 373	. . . . . . 7 |- ((A e. V /\ A =/= (/)) -> (f:A-onto->B -> (/) ~< B))
20		relen 4354	. . . . . . . . . 10 |- Rel ~~
21	20	brrelexi 3198	. . . . . . . . 9 |- (A ~~ n -> A e. V)
22	21	a1i 8	. . . . . . . 8 |- (n e. om -> (A ~~ n -> A e. V))
23	22	r19.23aiv 1735	. . . . . . 7 |- (E.n e. om A ~~ n -> A e. V)
24	19, 23	sylan 448	. . . . . 6 |- ((E.n e. om A ~~ n /\ A =/= (/)) -> (f:A-onto->B -> (/) ~< B))
25		fodomfi 4540	. . . . . . . 8 |- ((E.n e. om A ~~ n /\ f:A-onto->B) -> B ~<_ A)
26	25	ex 373	. . . . . . 7 |- (E.n e. om A ~~ n -> (f:A-onto->B -> B ~<_ A))
27	26	adantr 389	. . . . . 6 |- ((E.n e. om A ~~ n /\ A =/= (/)) -> (f:A-onto->B -> B ~<_ A))
28	24, 27	jcad 598	. . . . 5 |- ((E.n e. om A ~~ n /\ A =/= (/)) -> (f:A-onto->B -> ((/) ~< B /\ B ~<_ A)))
29	28	19.23adv 1209	. . . 4 |- ((E.n e. om A ~~ n /\ A =/= (/)) -> (E.f f:A-onto->B -> ((/) ~< B /\ B ~<_ A)))
30	29	ex 373	. . 3 |- (E.n e. om A ~~ n -> (A =/= (/) -> (E.f f:A-onto->B -> ((/) ~< B /\ B ~<_ A))))
31	30	imp3a 361	. 2 |- (E.n e. om A ~~ n -> ((A =/= (/) /\ E.f f:A-onto->B) -> ((/) ~< B /\ B ~<_ A)))
32		sdomdomtr 4449	. . . . 5 |- (A e. V -> (((/) ~< B /\ B ~<_ A) -> (/) ~< A))
33		0sdomg 4446	. . . . 5 |- (A e. V -> ((/) ~< A <-> A =/= (/)))
34	32, 33	sylibd 202	. . . 4 |- (A e. V -> (((/) ~< B /\ B ~<_ A) -> A =/= (/)))
35		fodomr 4463	. . . . 5 |- ((A e. V /\ (/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B)
36	35	3expib 834	. . . 4 |- (A e. V -> (((/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B))
37	34, 36	jcad 598	. . 3 |- (A e. V -> (((/) ~< B /\ B ~<_ A) -> (A =/= (/) /\ E.f f:A-onto->B)))
38	23, 37	syl 10	. 2 |- (E.n e. om A ~~ n -> (((/) ~< B /\ B ~<_ A) -> (A =/= (/) /\ E.f f:A-onto->B)))
39	31, 38	impbid 514	1 |- (E.n e. om A ~~ n -> ((A =/= (/) /\ E.f f:A-onto->B) <-> ((/) ~< B /\ B ~<_ A)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977   =/= wne 1577  E.wrex 1638  Vcvv 1802  (/)c0 2270   class class class wbr 2609  omcom 3121  dom cdm 3160  ran crn 3161  -->wf 3168  -onto->wfo 3170   ~~ cen 4348   ~<_ cdom 4349   ~< csdm 4350

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-1o 4117  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353

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