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Theorem fconst3 7164
Description: Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)
Assertion
Ref Expression
fconst3 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))

Proof of Theorem fconst3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fconstfv 7163 . 2 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))
2 fnfun 6603 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
3 fndm 6606 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4 eqimss2 4002 . . . . 5 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
53, 4syl 17 . . . 4 (𝐹 Fn 𝐴𝐴 ⊆ dom 𝐹)
6 funconstss 7007 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴 (𝐹𝑥) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
72, 5, 6syl2anc 585 . . 3 (𝐹 Fn 𝐴 → (∀𝑥𝐴 (𝐹𝑥) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
87pm5.32i 576 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
91, 8bitri 275 1 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wral 3065  wss 3911  {csn 4587  ccnv 5633  dom cdm 5634  cima 5637  Fun wfun 6491   Fn wfn 6492  wf 6493  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505
This theorem is referenced by:  fconst4  7165  dnsconst  22732
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