Theorem fodom 4808

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 4759. AC is not needed for finite sets - see fodomfi 4575.

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 3678	. . . 4 |- (F:A-onto->B -> F:A-->B)
2		fodom.1	. . . . 5 |- A e. V
3		fex 3658	. . . . 5 |- ((F:A-->B /\ A e. V) -> F e. V)
4	2, 3	mpan2 698	. . . 4 |- (F:A-->B -> F e. V)
5	1, 4	syl 10	. . 3 |- (F:A-onto->B -> F e. V)
6		cnvexg 3525	. . 3 |- (F e. V -> `'F e. V)
7		ac7g 4759	. . 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 3680	. . . . . . . 8 |- (F:A-onto->B -> ran F = B)
10		df-rn 3195	. . . . . . . 8 |- ran F = dom `' F
11	9, 10	syl5eqr 1524	. . . . . . 7 |- (F:A-onto->B -> dom `' F = B)
12		fneq2 3589	. . . . . . 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 4421	. . . . . . . 8 |- ((B ~<_ ran f /\ ran f ~<_ A) -> B ~<_ A)
15		fnfrn 3640	. . . . . . . . . . . . 13 |- (f Fn B <-> f:B-->ran f)
16	15	biimp 151	. . . . . . . . . . . 12 |- (f Fn B -> f:B-->ran f)
17	16	ad2antlr 407	. . . . . . . . . . 11 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> f:B-->ran f)
18		funss 3540	. . . . . . . . . . . . . 14 |- (`'f (_ F -> (Fun F -> Fun `'f))
19	18	impcom 351	. . . . . . . . . . . . 13 |- ((Fun F /\ `'f (_ F) -> Fun `'f)
20		fofun 3679	. . . . . . . . . . . . 13 |- (F:A-onto->B -> Fun F)
21		cnvss 3297	. . . . . . . . . . . . . 14 |- (f (_ `'F -> `'f (_ `'`'F)
22		cnvcnvss 3494	. . . . . . . . . . . . . . 15 |- `'`'F (_ F
23		sstr 2075	. . . . . . . . . . . . . . 15 |- ((`'f (_ `'`'F /\ `'`'F (_ F) -> `'f (_ F)
24	22, 23	mpan2 698	. . . . . . . . . . . . . 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 395	. . . . . . . . . . 11 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> Fun `'f)
28	17, 27	jca 288	. . . . . . . . . 10 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> (f:B-->ran f /\ Fun `'f))
29		df-f1 3201	. . . . . . . . . 10 |- (f:B-1-1->ran f <-> (f:B-->ran f /\ Fun `'f))
30	28, 29	sylibr 200	. . . . . . . . 9 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> f:B-1-1->ran f)
31		visset 1816	. . . . . . . . . . 11 |- f e. V
32	31	rnex 3367	. . . . . . . . . 10 |- ran f e. V
33		f1dom2g 4403	. . . . . . . . . 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 3348	. . . . . . . . . . . 12 |- (f (_ `'F -> ran f (_ ran `' F)
37	36	adantl 390	. . . . . . . . . . 11 |- ((F:A-onto->B /\ f (_ `'F) -> ran f (_ ran `' F)
38		fdm 3637	. . . . . . . . . . . . . 14 |- (F:A-->B -> dom F = A)
39	1, 38	syl 10	. . . . . . . . . . . . 13 |- (F:A-onto->B -> dom F = A)
40		dfdm4 3311	. . . . . . . . . . . . 13 |- dom F = ran `' F
41	39, 40	syl5eqr 1524	. . . . . . . . . . . 12 |- (F:A-onto->B -> ran `' F = A)
42	41	adantr 391	. . . . . . . . . . 11 |- ((F:A-onto->B /\ f (_ `'F) -> ran `' F = A)
43	37, 42	sseqtrd 2100	. . . . . . . . . 10 |- ((F:A-onto->B /\ f (_ `'F) -> ran f (_ A)
44		ssdomg 4414	. . . . . . . . . . 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 395	. . . . . . . 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 378	. . . . . 6 |- (F:A-onto->B -> (f Fn B -> (f (_ `'F -> B ~<_ A)))
50	13, 49	sylbid 203	. . . . 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 361	. . 3 |- (F:A-onto->B -> ((f (_ `'F /\ f Fn dom `' F) -> B ~<_ A))
53	52	19.23adv 1216	. 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 146   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  Vcvv 1814   (_ wss 2050   class class class wbr 2624  `'ccnv 3175  dom cdm 3176  ran crn 3177  Fun wfun 3182   Fn wfn 3183  -->wf 3184  -1-1->wf1 3185  -onto->wfo 3186   ~<_ cdom 4371

This theorem is referenced by:  fodomg 4809  fodomb 4810  brdom3 4811  brdom5 4812  brdom4 4813  qnnen 7504

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-en 4374  df-dom 4375

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