Theorem imadomg 3616

Description: An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92.

Assertion

Ref	Expression
imadomg	|- (A e. B -> (Fun F -> (F"A) ~<_ A))

Proof of Theorem imadomg

Step	Hyp	Ref	Expression
1		funres 2697	. . . . 5 |- (Fun F -> Fun (F |` A))
2		funforn 2792	. . . . 5 |- (Fun (F |` A) <-> (F |` A):dom (F |` A)-onto->ran (F |` A))
3	1, 2	sylib 173	. . . 4 |- (Fun F -> (F |` A):dom (F |` A)-onto->ran (F |` A))
4		resfunexg 2717	. . . . . . 7 |- (A e. B -> (Fun F -> (F |` A) e. V))
5		dmexg 2551	. . . . . . 7 |- ((F |` A) e. V -> dom (F |` A) e. V)
6	4, 5	syl6 23	. . . . . 6 |- (A e. B -> (Fun F -> dom (F |` A) e. V))
7		fodomg 3614	. . . . . . 7 |- (dom (F |` A) e. V -> ((F |` A):dom (F |` A)-onto->ran (F |` A) -> ran (F |` A) ~<_ dom (F |` A)))
8		df-ima 2431	. . . . . . . 8 |- (F"A) = ran (F |` A)
9	8	breq1i 2068	. . . . . . 7 |- ((F"A) ~<_ dom (F |` A) <-> ran (F |` A) ~<_ dom (F |` A))
10	7, 9	syl6ibr 186	. . . . . 6 |- (dom (F |` A) e. V -> ((F |` A):dom (F |` A)-onto->ran (F |` A) -> (F"A) ~<_ dom (F |` A)))
11	6, 10	syl6 23	. . . . 5 |- (A e. B -> (Fun F -> ((F |` A):dom (F |` A)-onto->ran (F |` A) -> (F"A) ~<_ dom (F |` A))))
12	11	com3l 34	. . . 4 |- (Fun F -> ((F |` A):dom (F |` A)-onto->ran (F |` A) -> (A e. B -> (F"A) ~<_ dom (F |` A))))
13	3, 12	mpd 46	. . 3 |- (Fun F -> (A e. B -> (F"A) ~<_ dom (F |` A)))
14	13	com12 13	. 2 |- (A e. B -> (Fun F -> (F"A) ~<_ dom (F |` A)))
15		domtr 3320	. . . . 5 |- (((F"A) ~<_ dom (F |` A) /\ dom (F |` A) ~<_ A) -> (F"A) ~<_ A)
16		dmres 2584	. . . . . . 7 |- dom (F |` A) = (A i^i dom F)
17		inss1 1657	. . . . . . 7 |- (A i^i dom F) (_ A
18	16, 17	eqsstr 1530	. . . . . 6 |- dom (F |` A) (_ A
19		ssdom2g 3312	. . . . . 6 |- (A e. B -> (dom (F |` A) (_ A -> dom (F |` A) ~<_ A))
20	18, 19	mpi 44	. . . . 5 |- (A e. B -> dom (F |` A) ~<_ A)
21	15, 20	sylan2 346	. . . 4 |- (((F"A) ~<_ dom (F |` A) /\ A e. B) -> (F"A) ~<_ A)
22	21	ancoms 334	. . 3 |- ((A e. B /\ (F"A) ~<_ dom (F |` A)) -> (F"A) ~<_ A)
23	22	exp 291	. 2 |- (A e. B -> ((F"A) ~<_ dom (F |` A) -> (F"A) ~<_ A))
24	14, 23	syld 27	1 |- (A e. B -> (Fun F -> (F"A) ~<_ A))

Syntax hints:   -> wi 2   e. wcel 1092  Vcvv 1348   i^i cin 1486   (_ wss 1487   class class class wbr 2054  dom cdm 2410  ran crn 2411   |` cres 2412  "cima 2413  Fun wfun 2416  -onto->wfo 2420   ~<_ cdom 3272

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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