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Theorem rrxf 24008
Description: Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.)
Hypotheses
Ref Expression
rrxmval.1 𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}
rrxf.1 (𝜑𝐹𝑋)
Assertion
Ref Expression
rrxf (𝜑𝐹:𝐼⟶ℝ)
Distinct variable groups:   ,𝐹   ,𝐼
Allowed substitution hints:   𝜑()   𝑋()

Proof of Theorem rrxf
StepHypRef Expression
1 rrxmval.1 . . . 4 𝑋 = { ∈ (ℝ ↑m 𝐼) ∣ finSupp 0}
21ssrab3 4043 . . 3 𝑋 ⊆ (ℝ ↑m 𝐼)
3 rrxf.1 . . 3 (𝜑𝐹𝑋)
42, 3sseldi 3951 . 2 (𝜑𝐹 ∈ (ℝ ↑m 𝐼))
5 elmapi 8424 . 2 (𝐹 ∈ (ℝ ↑m 𝐼) → 𝐹:𝐼⟶ℝ)
64, 5syl 17 1 (𝜑𝐹:𝐼⟶ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {crab 3137   class class class wbr 5052  wf 6339  (class class class)co 7149  m cmap 8402   finSupp cfsupp 8830  cr 10534  0cc0 10535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-map 8404
This theorem is referenced by:  rrxsuppss  24010  rrxmval  24012  rrxmetlem  24014  rrxmet  24015  rrxdstprj1  24016
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