Theorem fodom 4884

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 4835. AC is not needed for finite sets - see fodomfi 4650.

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 3747	. . . 4 |- (F:A-onto->B -> F:A-->B)
2		fodom.1	. . . . 5 |- A e. V
3		fex 3727	. . . . 5 |- ((F:A-->B /\ A e. V) -> F e. V)
4	2, 3	mpan2 699	. . . 4 |- (F:A-->B -> F e. V)
5	1, 4	syl 10	. . 3 |- (F:A-onto->B -> F e. V)
6		cnvexg 3594	. . 3 |- (F e. V -> `'F e. V)
7		ac7g 4835	. . 3 |- (`'F e. V -> E.f(f (_ `'F /\ f Fn dom `' F))
8	5, 6, 7	3syl 20	. 2 |- (F:A-onto->B -> E.f(f (_ `'F /\ f Fn dom `' F))
9		forn 3750	. . . . . . . 8 |- (F:A-onto->B -> ran F = B)
10		df-rn 3244	. . . . . . . 8 |- ran F = dom `' F
11	9, 10	syl5eqr 1558	. . . . . . 7 |- (F:A-onto->B -> dom `' F = B)
12		fneq2 3658	. . . . . . 7 |- (dom `' F = B -> (f Fn dom `' F <-> f Fn B))
13	11, 12	syl 10	. . . . . 6 |- (F:A-onto->B -> (f Fn dom `' F <-> f Fn B))
14		domtr 4502	. . . . . . . 8 |- ((B ~<_ ran f /\ ran f ~<_ A) -> B ~<_ A)
15		dffn3 3709	. . . . . . . . . . . . 13 |- (f Fn B <-> f:B-->ran f)
16	15	biimpi 149	. . . . . . . . . . . 12 |- (f Fn B -> f:B-->ran f)
17	16	ad2antlr 405	. . . . . . . . . . 11 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> f:B-->ran f)
18		funss 3609	. . . . . . . . . . . . . 14 |- (`'f (_ F -> (Fun F -> Fun `'f))
19	18	impcom 349	. . . . . . . . . . . . 13 |- ((Fun F /\ `'f (_ F) -> Fun `'f)
20		fofun 3748	. . . . . . . . . . . . 13 |- (F:A-onto->B -> Fun F)
21		cnvss 3354	. . . . . . . . . . . . . 14 |- (f (_ `'F -> `'f (_ `'`'F)
22		cnvcnvss 3543	. . . . . . . . . . . . . . 15 |- `'`'F (_ F
23		sstr 2116	. . . . . . . . . . . . . . 15 |- ((`'f (_ `'`'F /\ `'`'F (_ F) -> `'f (_ F)
24	22, 23	mpan2 699	. . . . . . . . . . . . . 14 |- (`'f (_ `'`'F -> `'f (_ F)
25	21, 24	syl 10	. . . . . . . . . . . . 13 |- (f (_ `'F -> `'f (_ F)
26	19, 20, 25	syl2an 456	. . . . . . . . . . . 12 |- ((F:A-onto->B /\ f (_ `'F) -> Fun `'f)
27	26	adantlr 393	. . . . . . . . . . 11 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> Fun `'f)
28	17, 27	jca 286	. . . . . . . . . 10 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> (f:B-->ran f /\ Fun `'f))
29		df-f1 3250	. . . . . . . . . 10 |- (f:B-1-1->ran f <-> (f:B-->ran f /\ Fun `'f))
30	28, 29	sylibr 198	. . . . . . . . 9 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> f:B-1-1->ran f)
31		visset 1851	. . . . . . . . . . 11 |- f e. V
32	31	rnex 3421	. . . . . . . . . 10 |- ran f e. V
33		f1dom2g 4484	. . . . . . . . . 10 |- (ran f e. V -> (f:B-1-1->ran f -> B ~<_ ran f))
34	32, 33	ax-mp 7	. . . . . . . . 9 |- (f:B-1-1->ran f -> B ~<_ ran f)
35	30, 34	syl 10	. . . . . . . 8 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> B ~<_ ran f)
36		rnss 3402	. . . . . . . . . . . 12 |- (f (_ `'F -> ran f (_ ran `' F)
37	36	adantl 388	. . . . . . . . . . 11 |- ((F:A-onto->B /\ f (_ `'F) -> ran f (_ ran `' F)
38		fdm 3706	. . . . . . . . . . . . . 14 |- (F:A-->B -> dom F = A)
39	1, 38	syl 10	. . . . . . . . . . . . 13 |- (F:A-onto->B -> dom F = A)
40		dfdm4 3369	. . . . . . . . . . . . 13 |- dom F = ran `' F
41	39, 40	syl5eqr 1558	. . . . . . . . . . . 12 |- (F:A-onto->B -> ran `' F = A)
42	41	adantr 389	. . . . . . . . . . 11 |- ((F:A-onto->B /\ f (_ `'F) -> ran `' F = A)
43	37, 42	sseqtrd 2141	. . . . . . . . . 10 |- ((F:A-onto->B /\ f (_ `'F) -> ran f (_ A)
44		ssdomg 4495	. . . . . . . . . . 11 |- (ran f e. V -> (ran f (_ A -> ran f ~<_ A))
45	32, 44	ax-mp 7	. . . . . . . . . 10 |- (ran f (_ A -> ran f ~<_ A)
46	43, 45	syl 10	. . . . . . . . 9 |- ((F:A-onto->B /\ f (_ `'F) -> ran f ~<_ A)
47	46	adantlr 393	. . . . . . . 8 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> ran f ~<_ A)
48	14, 35, 47	sylanc 473	. . . . . . 7 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> B ~<_ A)
49	48	exp31 376	. . . . . 6 |- (F:A-onto->B -> (f Fn B -> (f (_ `'F -> B ~<_ A)))
50	13, 49	sylbid 201	. . . . 5 |- (F:A-onto->B -> (f Fn dom `' F -> (f (_ `'F -> B ~<_ A)))
51	50	com23 32	. . . 4 |- (F:A-onto->B -> (f (_ `'F -> (f Fn dom `' F -> B ~<_ A)))
52	51	imp3a 359	. . 3 |- (F:A-onto->B -> ((f (_ `'F /\ f Fn dom `' F) -> B ~<_ A))
53	52	19.23adv 1247	. 2 |- (F:A-onto->B -> (E.f(f (_ `'F /\ f Fn dom `' F) -> B ~<_ A))
54	8, 53	mpd 26	1 |- (F:A-onto->B -> B ~<_ A)

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 988   e. wcel 990  E.wex 1012  Vcvv 1849   (_ wss 2091   class class class wbr 2669  `'ccnv 3224  dom cdm 3225  ran crn 3226  Fun wfun 3231   Fn wfn 3232  -->wf 3233  -1-1->wf1 3234  -onto->wfo 3235   ~<_ cdom 4452

This theorem is referenced by:  fodomg 4885  fodomb 4886  brdom3 4887  brdom5 4888  brdom4 4889  qnnen 7628

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-rep 2744  ax-sep 2754  ax-pow 2794  ax-pr 2832  ax-un 2920  ax-ac 4830

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-reu 1689  df-rab 1690  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252  df-fv 3253  df-en 4455  df-dom 4456

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