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Theorem sqrtrval 10564
Description: Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.)
Assertion
Ref Expression
sqrtrval (𝐴 ∈ ℝ → (√‘𝐴) = (𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem sqrtrval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2104 . . . 4 (𝑦 = 𝐴 → ((𝑥↑2) = 𝑦 ↔ (𝑥↑2) = 𝐴))
21anbi1d 454 . . 3 (𝑦 = 𝐴 → (((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥) ↔ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
32riotabidv 5648 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℝ ((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥)) = (𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
4 df-rsqrt 10562 . 2 √ = (𝑦 ∈ ℝ ↦ (𝑥 ∈ ℝ ((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥)))
5 reex 7573 . . 3 ℝ ∈ V
6 riotaexg 5650 . . 3 (ℝ ∈ V → (𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) ∈ V)
75, 6ax-mp 7 . 2 (𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) ∈ V
83, 4, 7fvmpt 5416 1 (𝐴 ∈ ℝ → (√‘𝐴) = (𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1296  wcel 1445  Vcvv 2633   class class class wbr 3867  cfv 5049  crio 5645  (class class class)co 5690  cr 7446  0cc0 7447  cle 7620  2c2 8571  cexp 10085  csqrt 10560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-cnex 7533  ax-resscn 7534
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-iota 5014  df-fun 5051  df-fv 5057  df-riota 5646  df-rsqrt 10562
This theorem is referenced by:  sqrt0  10568  resqrtcl  10593  rersqrtthlem  10594  sqrtsq  10608
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