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Theorem fconst3m 5646
Description: Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
Assertion
Ref Expression
fconst3m (∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem fconst3m
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fconstfvm 5645 . 2 (∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = 𝐵)))
2 fnfun 5227 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
3 fndm 5229 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4 eqimss2 3156 . . . . 5 (dom 𝐹 = 𝐴𝐴 ⊆ dom 𝐹)
53, 4syl 14 . . . 4 (𝐹 Fn 𝐴𝐴 ⊆ dom 𝐹)
6 funconstss 5545 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑦𝐴 (𝐹𝑦) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
72, 5, 6syl2anc 409 . . 3 (𝐹 Fn 𝐴 → (∀𝑦𝐴 (𝐹𝑦) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
87pm5.32i 450 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = 𝐵) ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))
91, 8syl6bb 195 1 (∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wex 1469  wcel 1481  wral 2417  wss 3075  {csn 3531  ccnv 4545  dom cdm 4546  cima 4549  Fun wfun 5124   Fn wfn 5125  wf 5126  cfv 5130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fo 5136  df-fv 5138
This theorem is referenced by:  fconst4m  5647
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