Theorem funfvima 7207

Description: A function's value in a preimage belongs to the image. (Contributed by NM, 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 5986	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
2	1	elin2 4169	. . . . . 6 ⊢ (𝐵 ∈ dom (𝐹 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹))
3		funres 6561	. . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
4		fvelrn 7051	. . . . . . . . 9 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ 𝐵 ∈ dom (𝐹 ↾ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴))
5	3, 4	sylan 580	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom (𝐹 ↾ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴))
6		df-ima 5654	. . . . . . . . . 10 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
7	6	eleq2i 2821	. . . . . . . . 9 ⊢ ((𝐹‘𝐵) ∈ (𝐹 “ 𝐴) ↔ (𝐹‘𝐵) ∈ ran (𝐹 ↾ 𝐴))
8		fvres 6880	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝐵) = (𝐹‘𝐵))
9	8	eleq1d 2814	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐴 → (((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴) ↔ (𝐹‘𝐵) ∈ ran (𝐹 ↾ 𝐴)))
10	7, 9	bitr4id 290	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → ((𝐹‘𝐵) ∈ (𝐹 “ 𝐴) ↔ ((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴)))
11	5, 10	syl5ibrcom 247	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom (𝐹 ↾ 𝐴)) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))
12	11	ex 412	. . . . . 6 ⊢ (Fun 𝐹 → (𝐵 ∈ dom (𝐹 ↾ 𝐴) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))))
13	2, 12	biimtrrid 243	. . . . 5 ⊢ (Fun 𝐹 → ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))))
14	13	expd 415	. . . 4 ⊢ (Fun 𝐹 → (𝐵 ∈ 𝐴 → (𝐵 ∈ dom 𝐹 → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))))
15	14	com12 32	. . 3 ⊢ (𝐵 ∈ 𝐴 → (Fun 𝐹 → (𝐵 ∈ dom 𝐹 → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))))
16	15	impd 410	. 2 ⊢ (𝐵 ∈ 𝐴 → ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))))
17	16	pm2.43b 55	1 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  dom cdm 5641  ran crn 5642   ↾ cres 5643   “ cima 5644  Fun wfun 6508  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522

This theorem is referenced by:  funfvima2  7208  elovimad  7440  tz7.48-2  8413  tz9.12lem3  9749  djuun  9886  swrdwrdsymb  14634  lindff1  21736  txcnp  23514  c1liplem1  25908  pthdivtx  29664  htthlem  30853  tpr2rico  33909  brsiga  34180  erdszelem8  35192  relowlpssretop  37359  limsuppnfdlem  45706  limsupresxr  45771  liminfresxr  45772  liminfvalxr  45788