Theorem funfvima 5459

Description: A function's value in a preimage belongs to the image. (Contributed by set.mm contributors, 23-Sep-2003.)

Assertion

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

Proof of Theorem funfvima

Step	Hyp	Ref	Expression
1		dmres 4986	. . . . . . . 8 ⊢ dom (F ↾ A) = (A ∩ dom F)
2	1	eleq2i 2417	. . . . . . 7 ⊢ (B ∈ dom (F ↾ A) ↔ B ∈ (A ∩ dom F))
3		elin 3219	. . . . . . 7 ⊢ (B ∈ (A ∩ dom F) ↔ (B ∈ A ∧ B ∈ dom F))
4	2, 3	bitri 240	. . . . . 6 ⊢ (B ∈ dom (F ↾ A) ↔ (B ∈ A ∧ B ∈ dom F))
5		funres 5143	. . . . . . . . 9 ⊢ (Fun F → Fun (F ↾ A))
6		fvelrn 5413	. . . . . . . . 9 ⊢ ((Fun (F ↾ A) ∧ B ∈ dom (F ↾ A)) → ((F ↾ A) ‘B) ∈ ran (F ↾ A))
7	5, 6	sylan 457	. . . . . . . 8 ⊢ ((Fun F ∧ B ∈ dom (F ↾ A)) → ((F ↾ A) ‘B) ∈ ran (F ↾ A))
8		fvres 5342	. . . . . . . . . 10 ⊢ (B ∈ A → ((F ↾ A) ‘B) = (F ‘B))
9	8	eleq1d 2419	. . . . . . . . 9 ⊢ (B ∈ A → (((F ↾ A) ‘B) ∈ ran (F ↾ A) ↔ (F ‘B) ∈ ran (F ↾ A)))
10		dfima3 4951	. . . . . . . . . 10 ⊢ (F “ A) = ran (F ↾ A)
11	10	eleq2i 2417	. . . . . . . . 9 ⊢ ((F ‘B) ∈ (F “ A) ↔ (F ‘B) ∈ ran (F ↾ A))
12	9, 11	syl6rbbr 255	. . . . . . . 8 ⊢ (B ∈ A → ((F ‘B) ∈ (F “ A) ↔ ((F ↾ A) ‘B) ∈ ran (F ↾ A)))
13	7, 12	syl5ibrcom 213	. . . . . . 7 ⊢ ((Fun F ∧ B ∈ dom (F ↾ A)) → (B ∈ A → (F ‘B) ∈ (F “ A)))
14	13	ex 423	. . . . . 6 ⊢ (Fun F → (B ∈ dom (F ↾ A) → (B ∈ A → (F ‘B) ∈ (F “ A))))
15	4, 14	syl5bir 209	. . . . 5 ⊢ (Fun F → ((B ∈ A ∧ B ∈ dom F) → (B ∈ A → (F ‘B) ∈ (F “ A))))
16	15	exp3a 425	. . . 4 ⊢ (Fun F → (B ∈ A → (B ∈ dom F → (B ∈ A → (F ‘B) ∈ (F “ A)))))
17	16	com12 27	. . 3 ⊢ (B ∈ A → (Fun F → (B ∈ dom F → (B ∈ A → (F ‘B) ∈ (F “ A)))))
18	17	imp3a 420	. 2 ⊢ (B ∈ A → ((Fun F ∧ B ∈ dom F) → (B ∈ A → (F ‘B) ∈ (F “ A))))
19	18	pm2.43b 46	1 ⊢ ((Fun F ∧ B ∈ dom F) → (B ∈ A → (F ‘B) ∈ (F “ A)))

Syntax hints:   → wi 4   ∧ wa 358   ∈ wcel 1710   ∩ cin 3208   “ cima 4722  dom cdm 4772  ran crn 4773   ↾ cres 4774  Fun wfun 4775   ‘cfv 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-fv 4795

This theorem is referenced by:  funfvima2  5460