Theorem fodom 5960

Description: An onto function implies dominance of domain over range. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 5911. AC is not needed for finite sets - see fodomfi 5656.

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		fex 4595	. . . 4 |- ((F:A-->B /\ A e. _V) -> F e. _V)
2		fof 4617	. . . 4 |- (F:A-onto->B -> F:A-->B)
3		fodom.1	. . . 4 |- A e. _V
4	1, 2, 3	sylancl 525	. . 3 |- (F:A-onto->B -> F e. _V)
5		cnvexg 4424	. . 3 |- (F e. _V -> `'F e. _V)
6		ac7g 5911	. . 3 |- (`'F e. _V -> E.f(f C_ `'F /\ f Fn dom `' F))
7	4, 5, 6	3syl 24	. 2 |- (F:A-onto->B -> E.f(f C_ `'F /\ f Fn dom `' F))
8		forn 4620	. . . . . . . 8 |- (F:A-onto->B -> ran F = B)
9		df-rn 4005	. . . . . . . 8 |- ran F = dom `' F
10	8, 9	syl5eqr 1942	. . . . . . 7 |- (F:A-onto->B -> dom `' F = B)
11	10	fneq2d 4506	. . . . . 6 |- (F:A-onto->B -> (f Fn dom `' F <-> f Fn B))
12		df-f1 4011	. . . . . . . . . 10 |- (f:B-1-1->ran f <-> (f:B-->ran f /\ Fun `'f))
13		dffn3 4570	. . . . . . . . . . . 12 |- (f Fn B <-> f:B-->ran f)
14	13	biimpi 168	. . . . . . . . . . 11 |- (f Fn B -> f:B-->ran f)
15	14	ad2antlr 441	. . . . . . . . . 10 |- (((F:A-onto->B /\ f Fn B) /\ f C_ `'F) -> f:B-->ran f)
16		funss 4439	. . . . . . . . . . . . 13 |- (`'f C_ F -> (Fun F -> Fun `'f))
17	16	impcom 378	. . . . . . . . . . . 12 |- ((Fun F /\ `'f C_ F) -> Fun `'f)
18		fofun 4618	. . . . . . . . . . . 12 |- (F:A-onto->B -> Fun F)
19		sstr 2625	. . . . . . . . . . . . 13 |- ((`'f C_ `'`'F /\ `'`'F C_ F) -> `'f C_ F)
20		cnvss 4134	. . . . . . . . . . . . 13 |- (f C_ `'F -> `'f C_ `'`'F)
21		cnvcnvss 4361	. . . . . . . . . . . . 13 |- `'`'F C_ F
22	19, 20, 21	sylancl 525	. . . . . . . . . . . 12 |- (f C_ `'F -> `'f C_ F)
23	17, 18, 22	syl2an 503	. . . . . . . . . . 11 |- ((F:A-onto->B /\ f C_ `'F) -> Fun `'f)
24	23	adantlr 429	. . . . . . . . . 10 |- (((F:A-onto->B /\ f Fn B) /\ f C_ `'F) -> Fun `'f)
25	12, 15, 24	sylanbrc 527	. . . . . . . . 9 |- (((F:A-onto->B /\ f Fn B) /\ f C_ `'F) -> f:B-1-1->ran f)
26		visset 2295	. . . . . . . . . . 11 |- f e. _V
27	26	rnex 4209	. . . . . . . . . 10 |- ran f e. _V
28		f1dom2g 5456	. . . . . . . . . 10 |- (ran f e. _V -> (f:B-1-1->ran f -> B ~<_ ran f))
29	27, 28	ax-mp 7	. . . . . . . . 9 |- (f:B-1-1->ran f -> B ~<_ ran f)
30	25, 29	syl 12	. . . . . . . 8 |- (((F:A-onto->B /\ f Fn B) /\ f C_ `'F) -> B ~<_ ran f)
31		rnss 4189	. . . . . . . . . . . 12 |- (f C_ `'F -> ran f C_ ran `' F)
32	31	adantl 424	. . . . . . . . . . 11 |- ((F:A-onto->B /\ f C_ `'F) -> ran f C_ ran `' F)
33		fdm 4567	. . . . . . . . . . . . . 14 |- (F:A-->B -> dom F = A)
34	2, 33	syl 12	. . . . . . . . . . . . 13 |- (F:A-onto->B -> dom F = A)
35		dfdm4 4151	. . . . . . . . . . . . 13 |- dom F = ran `' F
36	34, 35	syl5eqr 1942	. . . . . . . . . . . 12 |- (F:A-onto->B -> ran `' F = A)
37	36	adantr 425	. . . . . . . . . . 11 |- ((F:A-onto->B /\ f C_ `'F) -> ran `' F = A)
38	32, 37	sseqtrd 2653	. . . . . . . . . 10 |- ((F:A-onto->B /\ f C_ `'F) -> ran f C_ A)
39		ssdomg 5467	. . . . . . . . . . 11 |- (ran f e. _V -> (ran f C_ A -> ran f ~<_ A))
40	27, 39	ax-mp 7	. . . . . . . . . 10 |- (ran f C_ A -> ran f ~<_ A)
41	38, 40	syl 12	. . . . . . . . 9 |- ((F:A-onto->B /\ f C_ `'F) -> ran f ~<_ A)
42	41	adantlr 429	. . . . . . . 8 |- (((F:A-onto->B /\ f Fn B) /\ f C_ `'F) -> ran f ~<_ A)
43		domtr 5474	. . . . . . . 8 |- ((B ~<_ ran f /\ ran f ~<_ A) -> B ~<_ A)
44	30, 42, 43	syl11anc 524	. . . . . . 7 |- (((F:A-onto->B /\ f Fn B) /\ f C_ `'F) -> B ~<_ A)
45	44	exp31 407	. . . . . 6 |- (F:A-onto->B -> (f Fn B -> (f C_ `'F -> B ~<_ A)))
46	11, 45	sylbid 220	. . . . 5 |- (F:A-onto->B -> (f Fn dom `' F -> (f C_ `'F -> B ~<_ A)))
47	46	com23 36	. . . 4 |- (F:A-onto->B -> (f C_ `'F -> (f Fn dom `' F -> B ~<_ A)))
48	47	imp3a 388	. . 3 |- (F:A-onto->B -> ((f C_ `'F /\ f Fn dom `' F) -> B ~<_ A))
49	48	19.23adv 1584	. 2 |- (F:A-onto->B -> (E.f(f C_ `'F /\ f Fn dom `' F) -> B ~<_ A))
50	7, 49	mpd 29	1 |- (F:A-onto->B -> B ~<_ A)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   C_ wss 2593   class class class wbr 3338  `'ccnv 3985  dom cdm 3986  ran crn 3987  Fun wfun 3992   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  -onto->wfo 3996   ~<_ cdom 5424

This theorem is referenced by:  fodomg 5961  fodomb 5962  brdom3 5963  brdom5 5964  brdom4 5965  qnnen 8772  fictb 15371  2ndcctbss 15478

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  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-en 5427  df-dom 5428

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