Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvmap Structured version   Visualization version   GIF version

Theorem fvmap 40129
Description: Function value for a member of a set exponentiation. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
Hypotheses
Ref Expression
fvmap.a (𝜑𝐴𝑉)
fvmap.b (𝜑𝐵𝑊)
fvmap.f (𝜑𝐹 ∈ (𝐴𝑚 𝐵))
fvmap.c (𝜑𝐶𝐵)
Assertion
Ref Expression
fvmap (𝜑 → (𝐹𝐶) ∈ 𝐴)

Proof of Theorem fvmap
StepHypRef Expression
1 id 22 . 2 (𝜑𝜑)
2 fvmap.c . 2 (𝜑𝐶𝐵)
3 fvmap.f . . . 4 (𝜑𝐹 ∈ (𝐴𝑚 𝐵))
4 fvmap.a . . . . 5 (𝜑𝐴𝑉)
5 fvmap.b . . . . 5 (𝜑𝐵𝑊)
6 elmapg 8106 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐹 ∈ (𝐴𝑚 𝐵) ↔ 𝐹:𝐵𝐴))
74, 5, 6syl2anc 580 . . . 4 (𝜑 → (𝐹 ∈ (𝐴𝑚 𝐵) ↔ 𝐹:𝐵𝐴))
83, 7mpbid 224 . . 3 (𝜑𝐹:𝐵𝐴)
98ffvelrnda 6583 . 2 ((𝜑𝐶𝐵) → (𝐹𝐶) ∈ 𝐴)
101, 2, 9syl2anc 580 1 (𝜑 → (𝐹𝐶) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wcel 2157  wf 6095  cfv 6099  (class class class)co 6876  𝑚 cmap 8093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-map 8095
This theorem is referenced by:  ssmapsn  40148  hoidmvle  41548
  Copyright terms: Public domain W3C validator