Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rrx2pyel Structured version   Visualization version   GIF version

Theorem rrx2pyel 42276
Description: The y-coordinate of a point in a real Euclidean space of dimension 2 is a real number. (Contributed by AV, 2-Feb-2023.)
Hypotheses
Ref Expression
rrx2px.i 𝐼 = {1, 2}
rrx2px.b 𝑃 = (ℝ ↑𝑚 𝐼)
Assertion
Ref Expression
rrx2pyel (𝑋𝑃 → (𝑋‘2) ∈ ℝ)

Proof of Theorem rrx2pyel
StepHypRef Expression
1 rrx2px.b . 2 𝑃 = (ℝ ↑𝑚 𝐼)
2 id 22 . 2 (𝑋𝑃𝑋𝑃)
3 2ex 11435 . . . . 5 2 ∈ V
43prid2 4518 . . . 4 2 ∈ {1, 2}
5 rrx2px.i . . . 4 𝐼 = {1, 2}
64, 5eleqtrri 2905 . . 3 2 ∈ 𝐼
76a1i 11 . 2 (𝑋𝑃 → 2 ∈ 𝐼)
81, 2, 7mapfvd 8164 1 (𝑋𝑃 → (𝑋‘2) ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  wcel 2164  {cpr 4401  cfv 6127  (class class class)co 6910  𝑚 cmap 8127  cr 10258  1c1 10260  2c2 11413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-1cn 10317  ax-addcl 10319
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-map 8129  df-2 11421
This theorem is referenced by:  rrx2vlinest  43309  rrx2linest  43310  2sphere  43315  2sphere0  43316  line2  43318  line2x  43320  line2y  43321  itsclc0  43327  itsclinecirc0  43328
  Copyright terms: Public domain W3C validator