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Theorem sqrtrval 10740
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 2127 . . . 4 (𝑦 = 𝐴 → ((𝑥↑2) = 𝑦 ↔ (𝑥↑2) = 𝐴))
21anbi1d 460 . . 3 (𝑦 = 𝐴 → (((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥) ↔ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
32riotabidv 5700 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℝ ((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥)) = (𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
4 df-rsqrt 10738 . 2 √ = (𝑦 ∈ ℝ ↦ (𝑥 ∈ ℝ ((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥)))
5 reex 7722 . . 3 ℝ ∈ V
6 riotaexg 5702 . . 3 (ℝ ∈ V → (𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) ∈ V)
75, 6ax-mp 5 . 2 (𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) ∈ V
83, 4, 7fvmpt 5466 1 (𝐴 ∈ ℝ → (√‘𝐴) = (𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  Vcvv 2660   class class class wbr 3899  cfv 5093  crio 5697  (class class class)co 5742  cr 7587  0cc0 7588  cle 7769  2c2 8739  cexp 10260  csqrt 10736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-cnex 7679  ax-resscn 7680
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-rsqrt 10738
This theorem is referenced by:  sqrt0  10744  resqrtcl  10769  rersqrtthlem  10770  sqrtsq  10784
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