Theorem fodom 4778

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 4729. AC is not needed for finite sets - see fodomfi 4546.

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 3663	. . . 4 |- (F:A-onto->B -> F:A-->B)
2		fodom.1	. . . . 5 |- A e. V
3		fex 3643	. . . . 5 |- ((F:A-->B /\ A e. V) -> F e. V)
4	2, 3	mpan2 695	. . . 4 |- (F:A-->B -> F e. V)
5	1, 4	syl 10	. . 3 |- (F:A-onto->B -> F e. V)
6		cnvexg 3511	. . 3 |- (F e. V -> `'F e. V)
7		ac7g 4729	. . 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 3665	. . . . . . . 8 |- (F:A-onto->B -> ran F = B)
10		df-rn 3184	. . . . . . . 8 |- ran F = dom `' F
11	9, 10	syl5eqr 1518	. . . . . . 7 |- (F:A-onto->B -> dom `' F = B)
12		fneq2 3575	. . . . . . 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 4402	. . . . . . . 8 |- ((B ~<_ ran f /\ ran f ~<_ A) -> B ~<_ A)
15		fnfrn 3625	. . . . . . . . . . . . 13 |- (f Fn B <-> f:B-->ran f)
16	15	biimp 151	. . . . . . . . . . . 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 3526	. . . . . . . . . . . . . 14 |- (`'f (_ F -> (Fun F -> Fun `'f))
19	18	impcom 351	. . . . . . . . . . . . 13 |- ((Fun F /\ `'f (_ F) -> Fun `'f)
20		fofun 3664	. . . . . . . . . . . . 13 |- (F:A-onto->B -> Fun F)
21		cnvss 3286	. . . . . . . . . . . . . 14 |- (f (_ `'F -> `'f (_ `'`'F)
22		cnvcnvss 3480	. . . . . . . . . . . . . . 15 |- `'`'F (_ F
23		sstr 2068	. . . . . . . . . . . . . . 15 |- ((`'f (_ `'`'F /\ `'`'F (_ F) -> `'f (_ F)
24	22, 23	mpan2 695	. . . . . . . . . . . . . 14 |- (`'f (_ `'`'F -> `'f (_ F)
25	21, 24	syl 10	. . . . . . . . . . . . 13 |- (f (_ `'F -> `'f (_ F)
26	19, 20, 25	syl2an 454	. . . . . . . . . . . 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 288	. . . . . . . . . 10 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> (f:B-->ran f /\ Fun `'f))
29		df-f1 3190	. . . . . . . . . 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 1809	. . . . . . . . . . 11 |- f e. V
32		rnexg 3353	. . . . . . . . . . 11 |- (f e. V -> ran f e. V)
33	31, 32	ax-mp 7	. . . . . . . . . 10 |- ran f e. V
34		f1dom2g 4384	. . . . . . . . . 10 |- (ran f e. V -> (f:B-1-1->ran f -> B ~<_ ran f))
35	33, 34	ax-mp 7	. . . . . . . . 9 |- (f:B-1-1->ran f -> B ~<_ ran f)
36	30, 35	syl 10	. . . . . . . 8 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> B ~<_ ran f)
37		rnss 3337	. . . . . . . . . . . 12 |- (f (_ `'F -> ran f (_ ran `' F)
38	37	adantl 388	. . . . . . . . . . 11 |- ((F:A-onto->B /\ f (_ `'F) -> ran f (_ ran `' F)
39		fdm 3623	. . . . . . . . . . . . . 14 |- (F:A-->B -> dom F = A)
40	1, 39	syl 10	. . . . . . . . . . . . 13 |- (F:A-onto->B -> dom F = A)
41		dfdm4 3300	. . . . . . . . . . . . 13 |- dom F = ran `' F
42	40, 41	syl5eqr 1518	. . . . . . . . . . . 12 |- (F:A-onto->B -> ran `' F = A)
43	42	adantr 389	. . . . . . . . . . 11 |- ((F:A-onto->B /\ f (_ `'F) -> ran `' F = A)
44	38, 43	sseqtrd 2093	. . . . . . . . . 10 |- ((F:A-onto->B /\ f (_ `'F) -> ran f (_ A)
45		ssdomg 4395	. . . . . . . . . . 11 |- (ran f e. V -> (ran f (_ A -> ran f ~<_ A))
46	33, 45	ax-mp 7	. . . . . . . . . 10 |- (ran f (_ A -> ran f ~<_ A)
47	44, 46	syl 10	. . . . . . . . 9 |- ((F:A-onto->B /\ f (_ `'F) -> ran f ~<_ A)
48	47	adantlr 393	. . . . . . . 8 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> ran f ~<_ A)
49	14, 36, 48	sylanc 471	. . . . . . 7 |- (((F:A-onto->B /\ f Fn B) /\ f (_ `'F) -> B ~<_ A)
50	49	exp31 376	. . . . . 6 |- (F:A-onto->B -> (f Fn B -> (f (_ `'F -> B ~<_ A)))
51	13, 50	sylbid 203	. . . . 5 |- (F:A-onto->B -> (f Fn dom `' F -> (f (_ `'F -> B ~<_ A)))
52	51	com23 32	. . . 4 |- (F:A-onto->B -> (f (_ `'F -> (f Fn dom `' F -> B ~<_ A)))
53	52	imp3a 361	. . 3 |- (F:A-onto->B -> ((f (_ `'F /\ f Fn dom `' F) -> B ~<_ A))
54	53	19.23adv 1212	. 2 |- (F:A-onto->B -> (E.f(f (_ `'F /\ f Fn dom `' F) -> B ~<_ A))
55	8, 54	mpd 26	1 |- (F:A-onto->B -> B ~<_ A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  Vcvv 1807   (_ wss 2043   class class class wbr 2614  `'ccnv 3164  dom cdm 3165  ran crn 3166  Fun wfun 3171   Fn wfn 3172  -->wf 3173  -1-1->wf1 3174  -onto->wfo 3175   ~<_ cdom 4355

This theorem is referenced by:  fodomg 4779  fodomb 4780  brdom3 4781  brdom5 4782  brdom4 4783  qnnen 7454

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-pow 2737  ax-pr 2774  ax-un 2861  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-reu 1648  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  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-en 4357  df-dom 4358

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