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Theorem fconst3 7209
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 7208 . 2 (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))
2 fnfun 6642 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
3 fndm 6645 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4 eqimss2 4036 . . . . 5 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
53, 4syl 17 . . . 4 (𝐹 Fn 𝐴𝐴 ⊆ dom 𝐹)
6 funconstss 7050 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴 (𝐹𝑥) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
72, 5, 6syl2anc 583 . . 3 (𝐹 Fn 𝐴 → (∀𝑥𝐴 (𝐹𝑥) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
87pm5.32i 574 . 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 395   = wceq 1533  wral 3055  wss 3943  {csn 4623  ccnv 5668  dom cdm 5669  cima 5672  Fun wfun 6530   Fn wfn 6531  wf 6532  cfv 6536
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544
This theorem is referenced by:  fconst4  7210  dnsconst  23233
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