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Theorem rrxf 25328
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 4078 . . 3 𝑋 βŠ† (ℝ ↑m 𝐼)
3 rrxf.1 . . 3 (πœ‘ β†’ 𝐹 ∈ 𝑋)
42, 3sselid 3978 . 2 (πœ‘ β†’ 𝐹 ∈ (ℝ ↑m 𝐼))
5 elmapi 8867 . 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 1534   ∈ wcel 2099  {crab 3429   class class class wbr 5148  βŸΆwf 6544  (class class class)co 7420   ↑m cmap 8844   finSupp cfsupp 9385  β„cr 11137  0cc0 11138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-map 8846
This theorem is referenced by:  rrxsuppss  25330  rrxmval  25332  rrxmetlem  25334  rrxmet  25335  rrxdstprj1  25336
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