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Theorem mptelee 28420
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 28419 . 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 2730 . . . . 5 (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) = (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡))
42, 3fnmpti 6692 . . . 4 (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) Fn (1...𝑁)
5 df-f 6546 . . . 4 ((π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)):(1...𝑁)βŸΆβ„ ↔ ((π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) Fn (1...𝑁) ∧ ran (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) βŠ† ℝ))
64, 5mpbiran 705 . . 3 ((π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)):(1...𝑁)βŸΆβ„ ↔ ran (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) βŠ† ℝ)
73rnmpt 5953 . . . . 5 ran (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) = {π‘Ž ∣ βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡)}
87sseq1i 4009 . . . 4 (ran (π‘˜ ∈ (1...𝑁) ↦ (𝐴𝐹𝐡)) βŠ† ℝ ↔ {π‘Ž ∣ βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡)} βŠ† ℝ)
9 abss 4056 . . . . 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 3280 . . . . . . . . 9 β„²π‘˜βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡)
11 nfv 1915 . . . . . . . . 9 β„²π‘˜ π‘Ž ∈ ℝ
1210, 11nfim 1897 . . . . . . . 8 β„²π‘˜(βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ)
1312nfal 2314 . . . . . . 7 β„²π‘˜βˆ€π‘Ž(βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ)
14 r19.23v 3180 . . . . . . . . 9 (βˆ€π‘˜ ∈ (1...𝑁)(π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ) ↔ (βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ))
1514albii 1819 . . . . . . . 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 3281 . . . . . . . . 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 3242 . . . . . . . . . 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 3648 . . . . . . . . . 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 3252 . . . . . 6 (βˆ€π‘Ž(βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ) β†’ βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ)
23 nfra1 3279 . . . . . . . 8 β„²π‘˜βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ
24 rsp 3242 . . . . . . . . 9 (βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ β†’ (π‘˜ ∈ (1...𝑁) β†’ (𝐴𝐹𝐡) ∈ ℝ))
25 eleq1a 2826 . . . . . . . . 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 3261 . . . . . . 7 (βˆ€π‘˜ ∈ (1...𝑁)(𝐴𝐹𝐡) ∈ ℝ β†’ (βˆƒπ‘˜ ∈ (1...𝑁)π‘Ž = (𝐴𝐹𝐡) β†’ π‘Ž ∈ ℝ))
2827alrimiv 1928 . . . . . 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 1537   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆ€wral 3059  βˆƒwrex 3068   βŠ† wss 3947   ↦ cmpt 5230  ran crn 5676   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  β„cr 11111  1c1 11113  β„•cn 12216  ...cfz 13488  π”Όcee 28413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-ee 28416
This theorem is referenced by:  eleesub  28436  eleesubd  28437  axsegconlem1  28442  axsegconlem8  28449  axpasch  28466  axeuclidlem  28487  axcontlem2  28490
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