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Theorem rrxf 25273
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 4073 . . 3 𝑋 βŠ† (ℝ ↑m 𝐼)
3 rrxf.1 . . 3 (πœ‘ β†’ 𝐹 ∈ 𝑋)
42, 3sselid 3973 . 2 (πœ‘ β†’ 𝐹 ∈ (ℝ ↑m 𝐼))
5 elmapi 8840 . 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 1533   ∈ wcel 2098  {crab 3424   class class class wbr 5139  βŸΆwf 6530  (class class class)co 7402   ↑m cmap 8817   finSupp cfsupp 9358  β„cr 11106  0cc0 11107
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 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-map 8819
This theorem is referenced by:  rrxsuppss  25275  rrxmval  25277  rrxmetlem  25279  rrxmet  25280  rrxdstprj1  25281
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