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 ∈ 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 ∈ B → (Fun F → (F ↾ A) ∈ V))
5		dmexg 2551	. . . . . . 7 ⊢ ((F ↾ A) ∈ V → dom (F ↾ A) ∈ V)
6	4, 5	syl6 23	. . . . . 6 ⊢ (A ∈ B → (Fun F → dom (F ↾ A) ∈ V))
7		fodomg 3614	. . . . . . 7 ⊢ (dom (F ↾ A) ∈ 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) ∈ V → ((F ↾ A):dom (F ↾ A)–onto→ran (F ↾ A) → (F “ A) ≼ dom (F ↾ A)))
11	6, 10	syl6 23	. . . . 5 ⊢ (A ∈ 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 ∈ B → (F “ A) ≼ dom (F ↾ A))))
13	3, 12	mpd 46	. . 3 ⊢ (Fun F → (A ∈ B → (F “ A) ≼ dom (F ↾ A)))
14	13	com12 13	. 2 ⊢ (A ∈ 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 ∩ dom F)
17		inss1 1657	. . . . . . 7 ⊢ (A ∩ dom F) ⊆ A
18	16, 17	eqsstr 1530	. . . . . 6 ⊢ dom (F ↾ A) ⊆ A
19		ssdom2g 3312	. . . . . 6 ⊢ (A ∈ B → (dom (F ↾ A) ⊆ A → dom (F ↾ A) ≼ A))
20	18, 19	mpi 44	. . . . 5 ⊢ (A ∈ B → dom (F ↾ A) ≼ A)
21	15, 20	sylan2 346	. . . 4 ⊢ (((F “ A) ≼ dom (F ↾ A) ∧ A ∈ B) → (F “ A) ≼ A)
22	21	ancoms 334	. . 3 ⊢ ((A ∈ B ∧ (F “ A) ≼ dom (F ↾ A)) → (F “ A) ≼ A)
23	22	exp 291	. 2 ⊢ (A ∈ B → ((F “ A) ≼ dom (F ↾ A) → (F “ A) ≼ A))
24	14, 23	syld 27	1 ⊢ (A ∈ B → (Fun F → (F “ A) ≼ A))

Syntax hints:   → wi 2   ∈ wcel 1092  Vcvv 1348   ∩ 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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