MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frnssb Structured version   Visualization version   GIF version

Theorem frnssb 6556
Description: A function is a function into a subset of its codomain if all of its values are elements of this subset. (Contributed by AV, 7-Feb-2021.)
Assertion
Ref Expression
frnssb ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → (𝐹:𝐴𝑊𝐹:𝐴𝑉))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝑉
Allowed substitution hint:   𝑊(𝑘)

Proof of Theorem frnssb
StepHypRef Expression
1 simpr 479 . . . 4 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉)
2 ffn 6207 . . . 4 (𝐹:𝐴𝑊𝐹 Fn 𝐴)
31, 2anim12ci 592 . . 3 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑊) → (𝐹 Fn 𝐴 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉))
4 ffnfv 6553 . . 3 (𝐹:𝐴𝑉 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉))
53, 4sylibr 224 . 2 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑊) → 𝐹:𝐴𝑉)
6 simpl 474 . . . . 5 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → 𝑉𝑊)
76anim1i 593 . . . 4 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → (𝑉𝑊𝐹:𝐴𝑉))
87ancomd 466 . . 3 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → (𝐹:𝐴𝑉𝑉𝑊))
9 fss 6218 . . 3 ((𝐹:𝐴𝑉𝑉𝑊) → 𝐹:𝐴𝑊)
108, 9syl 17 . 2 (((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) ∧ 𝐹:𝐴𝑉) → 𝐹:𝐴𝑊)
115, 10impbida 913 1 ((𝑉𝑊 ∧ ∀𝑘𝐴 (𝐹𝑘) ∈ 𝑉) → (𝐹:𝐴𝑊𝐹:𝐴𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2140  wral 3051  wss 3716   Fn wfn 6045  wf 6046  cfv 6050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058
This theorem is referenced by:  wlkdlem1  26811
  Copyright terms: Public domain W3C validator