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

Theorem fconstfv 7218
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 7211. (Contributed by NM, 27-Aug-2004.) (Proof shortened by OpenAI, 25-Mar-2020.)
Assertion
Ref Expression
fconstfv (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem fconstfv
StepHypRef Expression
1 ffnfv 7122 . 2 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ {𝐵}))
2 fvex 6903 . . . . 5 (𝐹𝑥) ∈ V
32elsn 4638 . . . 4 ((𝐹𝑥) ∈ {𝐵} ↔ (𝐹𝑥) = 𝐵)
43ralbii 3083 . . 3 (∀𝑥𝐴 (𝐹𝑥) ∈ {𝐵} ↔ ∀𝑥𝐴 (𝐹𝑥) = 𝐵)
54anbi2i 621 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ {𝐵}) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))
61, 5bitri 274 1 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1534  wcel 2099  wral 3051  {csn 4623   Fn wfn 6538  wf 6539  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551
This theorem is referenced by:  fconst3  7219  repsdf2  14778  rrxcph  25405  lnon0  30725  df0op2  31679  matunitlindflem1  37327  poimir  37364  lfl1  38778
  Copyright terms: Public domain W3C validator