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Theorem fconstfv 7230
Description: A constant function expressed in terms of its functionality, domain, and value. See also fconst2 7223. (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 7134 . 2 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ {𝐵}))
2 fvex 6915 . . . . 5 (𝐹𝑥) ∈ V
32elsn 4647 . . . 4 ((𝐹𝑥) ∈ {𝐵} ↔ (𝐹𝑥) = 𝐵)
43ralbii 3090 . . 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 1533  wcel 2098  wral 3058  {csn 4632   Fn wfn 6548  wf 6549  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561
This theorem is referenced by:  fconst3  7231  repsdf2  14770  rrxcph  25348  lnon0  30636  df0op2  31590  matunitlindflem1  37130  poimir  37167  lfl1  38582
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