Theorem fodomfib 4547

Description: Equivalence of an onto mapping and dominance for a non-empty finite set. Unlike fodomb 4780 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 3663	. . . . . . . . . . . . . . 15 |- (f:A-onto->B -> f:A-->B)
2		fdm 3623	. . . . . . . . . . . . . . 15 |- (f:A-->B -> dom f = A)
3	1, 2	syl 10	. . . . . . . . . . . . . 14 |- (f:A-onto->B -> dom f = A)
4	3	eqeq1d 1480	. . . . . . . . . . . . 13 |- (f:A-onto->B -> (dom f = (/) <-> A = (/)))
5		forn 3665	. . . . . . . . . . . . . . 15 |- (f:A-onto->B -> ran f = B)
6	5	eqeq1d 1480	. . . . . . . . . . . . . 14 |- (f:A-onto->B -> (ran f = (/) <-> B = (/)))
7		dm0rn0 3325	. . . . . . . . . . . . . 14 |- (dom f = (/) <-> ran f = (/))
8	6, 7	syl5bb 531	. . . . . . . . . . . . 13 |- (f:A-onto->B -> (dom f = (/) <-> B = (/)))
9	4, 8	bitr3d 529	. . . . . . . . . . . 12 |- (f:A-onto->B -> (A = (/) <-> B = (/)))
10	9	necon3bid 1598	. . . . . . . . . . 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 3670	. . . . . . . . . . . 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 4452	. . . . . . . . . . 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 4360	. . . . . . . . . 10 |- Rel ~~
21	20	brrelexi 3203	. . . . . . . . 9 |- (A ~~ n -> A e. V)
22	21	a1i 8	. . . . . . . 8 |- (n e. om -> (A ~~ n -> A e. V))
23	22	r19.23aiv 1740	. . . . . . 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 4546	. . . . . . . 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 599	. . . . 5 |- ((E.n e. om A ~~ n /\ A =/= (/)) -> (f:A-onto->B -> ((/) ~< B /\ B ~<_ A)))
29	28	19.23adv 1212	. . . 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 4455	. . . . 5 |- (A e. V -> (((/) ~< B /\ B ~<_ A) -> (/) ~< A))
33		0sdomg 4452	. . . . 5 |- (A e. V -> ((/) ~< A <-> A =/= (/)))
34	32, 33	sylibd 202	. . . 4 |- (A e. V -> (((/) ~< B /\ B ~<_ A) -> A =/= (/)))
35		fodomr 4469	. . . . 5 |- ((A e. V /\ (/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B)
36	35	3expib 835	. . . 4 |- (A e. V -> (((/) ~< B /\ B ~<_ A) -> E.f f:A-onto->B))
37	34, 36	jcad 599	. . 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 515	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 954   e. wcel 956  E.wex 978   =/= wne 1582  E.wrex 1643  Vcvv 1807  (/)c0 2276   class class class wbr 2614  omcom 3126  dom cdm 3165  ran crn 3166  -->wf 3173  -onto->wfo 3175   ~~ cen 4354   ~<_ 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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  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-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  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-1o 4123  df-er 4251  df-en 4357  df-dom 4358  df-sdom 4359

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