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Theorem mptelee 28408
Description: A condition for a mapping to be an element of a Euclidean space. (Contributed by Scott Fenton, 7-Jun-2013.)
Assertion
Ref Expression
mptelee (𝑁 ∈ β„• β†’ ((π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) ∈ (π”Όβ€˜π‘) ↔ βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ))
Distinct variable group:   π‘˜,𝑁
Allowed substitution hints:   𝐴(π‘˜)   𝐡(π‘˜)   𝐹(π‘˜)

Proof of Theorem mptelee
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 elee 28407 . 2 (𝑁 ∈ β„• β†’ ((π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) ∈ (π”Όβ€˜π‘) ↔ (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)):(1...𝑁)βŸΆβ„))
2 ovex 7444 . . . . 5 (𝐴𝐹𝐡) ∈ V
3 eqid 2732 . . . . 5 (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) = (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡))
42, 3fnmpti 6693 . . . 4 (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) Fn (1...𝑁)
5 df-f 6547 . . . 4 ((π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)):(1...𝑁)βŸΆβ„ ↔ ((π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) Fn (1...𝑁) ∧ ran (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) βŠ† ℝ))
64, 5mpbiran 707 . . 3 ((π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)):(1...𝑁)βŸΆβ„ ↔ ran (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) βŠ† ℝ)
73rnmpt 5954 . . . . 5 ran (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) = {π‘Ž ∣ βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡)}
87sseq1i 4010 . . . 4 (ran (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) βŠ† ℝ ↔ {π‘Ž ∣ βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡)} βŠ† ℝ)
9 abss 4057 . . . . 5 ({π‘Ž ∣ βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡)} βŠ† ℝ ↔ βˆ€π‘Ž(βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ))
10 nfre1 3282 . . . . . . . . 9 β„²π‘˜βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡)
11 nfv 1917 . . . . . . . . 9 β„²π‘˜ π‘Ž ∈ ℝ
1210, 11nfim 1899 . . . . . . . 8 β„²π‘˜(βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ)
1312nfal 2316 . . . . . . 7 β„²π‘˜βˆ€π‘Ž(βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ)
14 r19.23v 3182 . . . . . . . . 9 (βˆ€π‘˜ ∈ (1...𝑁)(π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ) ↔ (βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ))
1514albii 1821 . . . . . . . 8 (βˆ€π‘Žβˆ€π‘˜ ∈ (1...𝑁)(π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ) ↔ βˆ€π‘Ž(βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ))
16 ralcom4 3283 . . . . . . . . 9 (βˆ€π‘˜ ∈ (1...𝑁)βˆ€π‘Ž(π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ) ↔ βˆ€π‘Žβˆ€π‘˜ ∈ (1...𝑁)(π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ))
17 rsp 3244 . . . . . . . . . 10 (βˆ€π‘˜ ∈ (1...𝑁)βˆ€π‘Ž(π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ) β†’ (π‘˜ ∈ (1...𝑁) β†’ βˆ€π‘Ž(π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ)))
182clel2 3649 . . . . . . . . . 10 ((𝐴𝐹𝐡) ∈ ℝ ↔ βˆ€π‘Ž(π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ))
1917, 18imbitrrdi 251 . . . . . . . . 9 (βˆ€π‘˜ ∈ (1...𝑁)βˆ€π‘Ž(π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ) β†’ (π‘˜ ∈ (1...𝑁) β†’ (𝐴𝐹𝐡) ∈ ℝ))
2016, 19sylbir 234 . . . . . . . 8 (βˆ€π‘Žβˆ€π‘˜ ∈ (1...𝑁)(π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ) β†’ (π‘˜ ∈ (1...𝑁) β†’ (𝐴𝐹𝐡) ∈ ℝ))
2115, 20sylbir 234 . . . . . . 7 (βˆ€π‘Ž(βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ) β†’ (π‘˜ ∈ (1...𝑁) β†’ (𝐴𝐹𝐡) ∈ ℝ))
2213, 21ralrimi 3254 . . . . . 6 (βˆ€π‘Ž(βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ) β†’ βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ)
23 nfra1 3281 . . . . . . . 8 β„²π‘˜βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ
24 rsp 3244 . . . . . . . . 9 (βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ β†’ (π‘˜ ∈ (1...𝑁) β†’ (𝐴𝐹𝐡) ∈ ℝ))
25 eleq1a 2828 . . . . . . . . 9 ((𝐴𝐹𝐡) ∈ ℝ β†’ (π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ))
2624, 25syl6 35 . . . . . . . 8 (βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ β†’ (π‘˜ ∈ (1...𝑁) β†’ (π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ)))
2723, 11, 26rexlimd 3263 . . . . . . 7 (βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ β†’ (βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ))
2827alrimiv 1930 . . . . . 6 (βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ β†’ βˆ€π‘Ž(βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ))
2922, 28impbii 208 . . . . 5 (βˆ€π‘Ž(βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ) ↔ βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ)
309, 29bitri 274 . . . 4 ({π‘Ž ∣ βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡)} βŠ† ℝ ↔ βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ)
318, 30bitri 274 . . 3 (ran (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) βŠ† ℝ ↔ βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ)
326, 31bitri 274 . 2 ((π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)):(1...𝑁)βŸΆβ„ ↔ βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ)
331, 32bitrdi 286 1 (𝑁 ∈ β„• β†’ ((π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) ∈ (π”Όβ€˜π‘) ↔ βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205  βˆ€wal 1539   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3948   ↦ cmpt 5231  ran crn 5677   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  β„cr 11111  1c1 11113  β„•cn 12216  ...cfz 13488  π”Όcee 28401
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-ee 28404
This theorem is referenced by:  eleesub  28424  eleesubd  28425  axsegconlem1  28430  axsegconlem8  28437  axpasch  28454  axeuclidlem  28475  axcontlem2  28478
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