Theorem imadomg 4806

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		fodomg 4799	. . . . 5 |- (dom ( F |` A) e. V -> ((F |` A):dom ( F |` A)-onto->ran ( F |` A) -> ran ( F |` A) ~<_ dom ( F |` A)))
2		resfunexg 3579	. . . . . 6 |- ((Fun F /\ A e. B) -> (F |` A) e. V)
3		dmexg 3358	. . . . . 6 |- ((F |` A) e. V -> dom ( F |` A) e. V)
4	2, 3	syl 10	. . . . 5 |- ((Fun F /\ A e. B) -> dom ( F |` A) e. V)
5		funres 3551	. . . . . . 7 |- (Fun F -> Fun (F |` A))
6		funforn 3678	. . . . . . 7 |- (Fun (F |` A) <-> (F |` A):dom ( F |` A)-onto->ran ( F |` A))
7	5, 6	sylib 198	. . . . . 6 |- (Fun F -> (F |` A):dom ( F |` A)-onto->ran ( F |` A))
8	7	adantr 389	. . . . 5 |- ((Fun F /\ A e. B) -> (F |` A):dom ( F |` A)-onto->ran ( F |` A))
9	1, 4, 8	sylc 68	. . . 4 |- ((Fun F /\ A e. B) -> ran ( F |` A) ~<_ dom ( F |` A))
10		df-ima 3191	. . . 4 |- (F"A) = ran ( F |` A)
11	9, 10	syl5eqbr 2648	. . 3 |- ((Fun F /\ A e. B) -> (F"A) ~<_ dom ( F |` A))
12	11	expcom 374	. 2 |- (A e. B -> (Fun F -> (F"A) ~<_ dom ( F |` A)))
13		domtr 4415	. . . 4 |- (((F"A) ~<_ dom ( F |` A) /\ dom ( F |` A) ~<_ A) -> (F"A) ~<_ A)
14		dmres 3380	. . . . . 6 |- dom ( F |` A) = (A i^i dom F)
15		inss1 2230	. . . . . 6 |- (A i^i dom F) (_ A
16	14, 15	eqsstr 2091	. . . . 5 |- dom ( F |` A) (_ A
17		ssdom2g 4409	. . . . 5 |- (A e. B -> (dom ( F |` A) (_ A -> dom ( F |` A) ~<_ A))
18	16, 17	mpi 44	. . . 4 |- (A e. B -> dom ( F |` A) ~<_ A)
19	13, 18	sylan2 451	. . 3 |- (((F"A) ~<_ dom ( F |` A) /\ A e. B) -> (F"A) ~<_ A)
20	19	expcom 374	. 2 |- (A e. B -> ((F"A) ~<_ dom ( F |` A) -> (F"A) ~<_ A))
21	12, 20	syld 27	1 |- (A e. B -> (Fun F -> (F"A) ~<_ A))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  Vcvv 1811   i^i cin 2046   (_ wss 2047   class class class wbr 2619  dom cdm 3170  ran crn 3171   |` cres 3172  "cima 3173  Fun wfun 3176  -onto->wfo 3180   ~<_ cdom 4365

This theorem is referenced by:  uniimadom 4810

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-en 4368  df-dom 4369

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