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Theorem fconst3 7229
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 7228 . 2 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))
2 fnfun 6657 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
3 fndm 6660 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4 eqimss2 4039 . . . . 5 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
53, 4syl 17 . . . 4 (𝐹 Fn 𝐴𝐴 ⊆ dom 𝐹)
6 funconstss 7068 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴 (𝐹𝑥) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
72, 5, 6syl2anc 582 . . 3 (𝐹 Fn 𝐴 → (∀𝑥𝐴 (𝐹𝑥) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
87pm5.32i 573 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵) ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
91, 8bitri 274 1 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wral 3057  wss 3947  {csn 4630  ccnv 5679  dom cdm 5680  cima 5683  Fun wfun 6545   Fn wfn 6546  wf 6547  cfv 6551
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 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559
This theorem is referenced by:  fconst4  7230  dnsconst  23300
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