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Theorem rrxf 24768
Description: Euclidean vectors as functions. (Contributed by Thierry Arnoux, 7-Jul-2019.)
Hypotheses
Ref Expression
rrxmval.1 𝑋 = {β„Ž ∈ (ℝ ↑m 𝐼) ∣ β„Ž finSupp 0}
rrxf.1 (πœ‘ β†’ 𝐹 ∈ 𝑋)
Assertion
Ref Expression
rrxf (πœ‘ β†’ 𝐹:πΌβŸΆβ„)
Distinct variable groups:   β„Ž,𝐹   β„Ž,𝐼
Allowed substitution hints:   πœ‘(β„Ž)   𝑋(β„Ž)

Proof of Theorem rrxf
StepHypRef Expression
1 rrxmval.1 . . . 4 𝑋 = {β„Ž ∈ (ℝ ↑m 𝐼) ∣ β„Ž finSupp 0}
21ssrab3 4041 . . 3 𝑋 βŠ† (ℝ ↑m 𝐼)
3 rrxf.1 . . 3 (πœ‘ β†’ 𝐹 ∈ 𝑋)
42, 3sselid 3943 . 2 (πœ‘ β†’ 𝐹 ∈ (ℝ ↑m 𝐼))
5 elmapi 8788 . 2 (𝐹 ∈ (ℝ ↑m 𝐼) β†’ 𝐹:πΌβŸΆβ„)
64, 5syl 17 1 (πœ‘ β†’ 𝐹:πΌβŸΆβ„)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3408   class class class wbr 5106  βŸΆwf 6493  (class class class)co 7358   ↑m cmap 8766   finSupp cfsupp 9306  β„cr 11051  0cc0 11052
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-pow 5321  ax-pr 5385  ax-un 7673
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-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  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  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8768
This theorem is referenced by:  rrxsuppss  24770  rrxmval  24772  rrxmetlem  24774  rrxmet  24775  rrxdstprj1  24776
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