Theorem fodom 3613

Description: An onto function implies dominance of range over domain. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 3570.

Hypothesis

Ref	Expression
fodom.1	|- A e. V

Assertion

Ref	Expression
fodom	|- (F:A-onto->B -> B ~<_ A)

Proof of Theorem fodom

Step	Hyp	Ref	Expression
1		fof 2788	. . . 4 |- (F:A-onto->B -> F:A-->B)
2		fodom.1	. . . . 5 |- A e. V
3		fex 2771	. . . . 5 |- (A e. V -> (F:A-->B -> F e. V))
4	2, 3	ax-mp 6	. . . 4 |- (F:A-->B -> F e. V)
5	1, 4	syl 12	. . 3 |- (F:A-onto->B -> F e. V)
6		cnvexg 2669	. . 3 |- (F e. V -> `'F e. V)
7		ac7g 3570	. . 3 |- (`'F e. V -> E.f(f (_ `'F /\ f Fn dom `'F))
8	5, 6, 7	3syl 21	. 2 |- (F:A-onto->B -> E.f(f (_ `'F /\ f Fn dom `'F))
9		forn 2789	. . . . . . . 8 |- (F:A-onto->B -> ran F = B)
10		df-rn 2429	. . . . . . . 8 |- ran F = dom `'F
11	9, 10	syl5eqr 1138	. . . . . . 7 |- (F:A-onto->B -> dom `'F = B)
12		fneq2 2719	. . . . . . 7 |- (dom `'F = B -> (f Fn dom `'F <-> f Fn B))
13	11, 12	syl 12	. . . . . 6 |- (F:A-onto->B -> (f Fn dom `'F <-> f Fn B))
14		domtr 3320	. . . . . . . 8 |- ((B ~<_ ran f /\ ran f ~<_ A) -> B ~<_ A)
15		fnfrn 2758	. . . . . . . . . . . . 13 |- (f Fn B <-> f:B-->ran f)
16	15	biimp 133	. . . . . . . . . . . 12 |- (f Fn B -> f:B-->ran f)
17	16	ad2antlr 321	. . . . . . . . . . 11 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> f:B-->ran f)
18		funss 2682	. . . . . . . . . . . . . . 15 |- (`'f (_ F -> (Fun F -> Fun `'f))
19	18	com12 13	. . . . . . . . . . . . . 14 |- (Fun F -> (`'f (_ F -> Fun `'f))
20	19	imp 277	. . . . . . . . . . . . 13 |- ((Fun F /\ `'f (_ F) -> Fun `'f)
21		ffun 2754	. . . . . . . . . . . . . 14 |- (F:A-->B -> Fun F)
22	1, 21	syl 12	. . . . . . . . . . . . 13 |- (F:A-onto->B -> Fun F)
23		cnvss 2512	. . . . . . . . . . . . . 14 |- (f (_ `'F -> `'f (_ `'`'F)
24		cnvcnvss 2662	. . . . . . . . . . . . . . 15 |- `'`'F (_ F
25		sstr 1511	. . . . . . . . . . . . . . 15 |- ((`'f (_ `'`'F /\ `'`'F (_ F) -> `'f (_ F)
26	24, 25	mpan2 519	. . . . . . . . . . . . . 14 |- (`'f (_ `'`'F -> `'f (_ F)
27	23, 26	syl 12	. . . . . . . . . . . . 13 |- (f (_ `'F -> `'f (_ F)
28	20, 22, 27	syl2an 349	. . . . . . . . . . . 12 |- ((F:A-onto->B /\ f (_ `'F) -> Fun `'f)
29	28	adantlr 310	. . . . . . . . . . 11 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> Fun `'f)
30	17, 29	jca 236	. . . . . . . . . 10 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> (f:B-->ran f /\ Fun `'f))
31		df-f1 2435	. . . . . . . . . 10 |- (f:B-1-1->ran f <-> (f:B-->ran f /\ Fun `'f))
32	30, 31	sylibr 175	. . . . . . . . 9 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> f:B-1-1->ran f)
33		visset 1350	. . . . . . . . . . 11 |- f e. V
34		rnexg 2569	. . . . . . . . . . 11 |- (f e. V -> ran f e. V)
35	33, 34	ax-mp 6	. . . . . . . . . 10 |- ran f e. V
36		f1dom2g 3300	. . . . . . . . . 10 |- (ran f e. V -> (f:B-1-1->ran f -> B ~<_ ran f))
37	35, 36	ax-mp 6	. . . . . . . . 9 |- (f:B-1-1->ran f -> B ~<_ ran f)
38	32, 37	syl 12	. . . . . . . 8 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> B ~<_ ran f)
39		rnss 2558	. . . . . . . . . . . 12 |- (f (_ `'F -> ran f (_ ran `'F)
40	39	adantl 305	. . . . . . . . . . 11 |- ((F:A-onto->B /\ f (_ `'F) -> ran f (_ ran `'F)
41		fdm 2756	. . . . . . . . . . . . . 14 |- (F:A-->B -> dom F = A)
42	1, 41	syl 12	. . . . . . . . . . . . 13 |- (F:A-onto->B -> dom F = A)
43		dfdm4 2525	. . . . . . . . . . . . 13 |- dom F = ran `'F
44	42, 43	syl5eqr 1138	. . . . . . . . . . . 12 |- (F:A-onto->B -> ran `'F = A)
45	44	adantr 306	. . . . . . . . . . 11 |- ((F:A-onto->B /\ f (_ `'F) -> ran `'F = A)
46	40, 45	sseqtrd 1536	. . . . . . . . . 10 |- ((F:A-onto->B /\ f (_ `'F) -> ran f (_ A)
47		ssdomg 3311	. . . . . . . . . . 11 |- (ran f e. V -> (ran f (_ A -> ran f ~<_ A))
48	35, 47	ax-mp 6	. . . . . . . . . 10 |- (ran f (_ A -> ran f ~<_ A)
49	46, 48	syl 12	. . . . . . . . 9 |- ((F:A-onto->B /\ f (_ `'F) -> ran f ~<_ A)
50	49	adantlr 310	. . . . . . . 8 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> ran f ~<_ A)
51	14, 38, 50	sylanc 361	. . . . . . 7 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> B ~<_ A)
52	51	exp31 293	. . . . . 6 |- (F:A-onto->B -> (f Fn B -> (f (_ `'F -> B ~<_ A)))
53	13, 52	sylbid 178	. . . . 5 |- (F:A-onto->B -> (f Fn dom `'F -> (f (_ `'F -> B ~<_ A)))
54	53	com23 32	. . . 4 |- (F:A-onto->B -> (f (_ `'F -> (f Fn dom `'F -> B ~<_ A)))
55	54	imp3a 279	. . 3 |- (F:A-onto->B -> ((f (_ `'F /\ f Fn dom `'F) -> B ~<_ A))
56	55	19.23adv 954	. 2 |- (F:A-onto->B -> (E.f(f (_ `'F /\ f Fn dom `'F) -> B ~<_ A))
57	8, 56	mpd 46	1 |- (F:A-onto->B -> B ~<_ A)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487   class class class wbr 2054  `'ccnv 2409  dom cdm 2410  ran crn 2411  Fun wfun 2416   Fn wfn 2417  -->wf 2418  -1-1->wf1 2419  -onto->wfo 2420   ~<_ cdom 3272

This theorem is referenced by:  fodomg 3614  fodomb 3615  qnnen 4931

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-fr 2169  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-en 3274  df-dom 3275

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