Theorem imadomg 4789

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 4782	. . . . 5 |- (dom ( F |` A) e. V -> ((F |` A):dom ( F |` A)-onto->ran ( F |` A) -> ran ( F |` A) ~<_ dom ( F |` A)))
2		resfunexg 3575	. . . . . 6 |- ((Fun F /\ A e. B) -> (F |` A) e. V)
3		dmexg 3354	. . . . . 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 3547	. . . . . . 7 |- (Fun F -> Fun (F |` A))
6		funforn 3673	. . . . . . 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 3187	. . . 4 |- (F"A) = ran ( F |` A)
11	9, 10	syl5eqbr 2644	. . 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 4405	. . . 4 |- (((F"A) ~<_ dom ( F |` A) /\ dom ( F |` A) ~<_ A) -> (F"A) ~<_ A)
14		dmres 3376	. . . . . 6 |- dom ( F |` A) = (A i^i dom F)
15		inss1 2227	. . . . . 6 |- (A i^i dom F) (_ A
16	14, 15	eqsstr 2088	. . . . 5 |- dom ( F |` A) (_ A
17		ssdom2g 4399	. . . . 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 957  Vcvv 1808   i^i cin 2043   (_ wss 2044   class class class wbr 2615  dom cdm 3166  ran crn 3167   |` cres 3168  "cima 3169  Fun wfun 3172  -onto->wfo 3176   ~<_ cdom 4358

This theorem is referenced by:  uniimadom 4793

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-en 4360  df-dom 4361

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