Theorem funfvima 2904

Description: A function's value in a pre-image belongs to the image.

Assertion

Ref	Expression
funfvima	⊢ ((Fun F ∧ B ∈ dom F) → (B ∈ A → (F ‘B) ∈ (F “ A)))

Proof of Theorem funfvima

Step	Hyp	Ref	Expression
1		fvres 2840	. . . . . . . . . . 11 ⊢ (B ∈ A → ((F ↾ A) ‘B) = (F ‘B))
2	1	eleq1d 1155	. . . . . . . . . 10 ⊢ (B ∈ A → (((F ↾ A) ‘B) ∈ ran (F ↾ A) ↔ (F ‘B) ∈ ran (F ↾ A)))
3		df-ima 2431	. . . . . . . . . . 11 ⊢ (F “ A) = ran (F ↾ A)
4	3	eleq2i 1153	. . . . . . . . . 10 ⊢ ((F ‘B) ∈ (F “ A) ↔ (F ‘B) ∈ ran (F ↾ A))
5	2, 4	syl6rbbr 417	. . . . . . . . 9 ⊢ (B ∈ A → ((F ‘B) ∈ (F “ A) ↔ ((F ↾ A) ‘B) ∈ ran (F ↾ A)))
6		fvrn 2888	. . . . . . . . . 10 ⊢ ((Fun (F ↾ A) ∧ B ∈ dom (F ↾ A)) → ((F ↾ A) ‘B) ∈ ran (F ↾ A))
7		funres 2697	. . . . . . . . . 10 ⊢ (Fun F → Fun (F ↾ A))
8	6, 7	sylan 343	. . . . . . . . 9 ⊢ ((Fun F ∧ B ∈ dom (F ↾ A)) → ((F ↾ A) ‘B) ∈ ran (F ↾ A))
9	5, 8	syl5bir 184	. . . . . . . 8 ⊢ (B ∈ A → ((Fun F ∧ B ∈ dom (F ↾ A)) → (F ‘B) ∈ (F “ A)))
10	9	com12 13	. . . . . . 7 ⊢ ((Fun F ∧ B ∈ dom (F ↾ A)) → (B ∈ A → (F ‘B) ∈ (F “ A)))
11	10	exp 291	. . . . . 6 ⊢ (Fun F → (B ∈ dom (F ↾ A) → (B ∈ A → (F ‘B) ∈ (F “ A))))
12		dmres 2584	. . . . . . . 8 ⊢ dom (F ↾ A) = (A ∩ dom F)
13	12	eleq2i 1153	. . . . . . 7 ⊢ (B ∈ dom (F ↾ A) ↔ B ∈ (A ∩ dom F))
14		elin 1635	. . . . . . 7 ⊢ (B ∈ (A ∩ dom F) ↔ (B ∈ A ∧ B ∈ dom F))
15	13, 14	bitr 151	. . . . . 6 ⊢ (B ∈ dom (F ↾ A) ↔ (B ∈ A ∧ B ∈ dom F))
16	11, 15	syl5ibr 182	. . . . 5 ⊢ (Fun F → ((B ∈ A ∧ B ∈ dom F) → (B ∈ A → (F ‘B) ∈ (F “ A))))
17	16	exp3a 292	. . . 4 ⊢ (Fun F → (B ∈ A → (B ∈ dom F → (B ∈ A → (F ‘B) ∈ (F “ A)))))
18	17	com12 13	. . 3 ⊢ (B ∈ A → (Fun F → (B ∈ dom F → (B ∈ A → (F ‘B) ∈ (F “ A)))))
19	18	imp3a 279	. 2 ⊢ (B ∈ A → ((Fun F ∧ B ∈ dom F) → (B ∈ A → (F ‘B) ∈ (F “ A))))
20	19	pm2.43b 61	1 ⊢ ((Fun F ∧ B ∈ dom F) → (B ∈ A → (F ‘B) ∈ (F “ A)))

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092   ∩ cin 1486  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413  Fun wfun 2416   ‘cfv 2422

This theorem is referenced by:  funfvima2 2905  isomin 2937  isofrlem 2939  tz9.12lem3 3505

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  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-id 2125  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-fv 2438

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