Theorem sqrtrval 11144

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.

Step	Hyp	Ref	Expression
1		eqeq2 2203	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥↑2) = 𝑦 ↔ (𝑥↑2) = 𝐴))
2	1	anbi1d 465	. . 3 ⊢ (𝑦 = 𝐴 → (((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥) ↔ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
3	2	riotabidv 5875	. 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥)) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
4		df-rsqrt 11142	. 2 ⊢ √ = (𝑦 ∈ ℝ ↦ (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥)))
5		reex 8006	. . 3 ⊢ ℝ ∈ V
6		riotaexg 5877	. . 3 ⊢ (ℝ ∈ V → (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) ∈ V)
7	5, 6	ax-mp 5	. 2 ⊢ (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) ∈ V
8	3, 4, 7	fvmpt 5634	1 ⊢ (𝐴 ∈ ℝ → (√‘𝐴) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760   class class class wbr 4029  ‘cfv 5254  ℩crio 5872  (class class class)co 5918  ℝcr 7871  0cc0 7872   ≤ cle 8055  2c2 9033  ↑cexp 10609  √csqrt 11140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-rsqrt 11142

This theorem is referenced by:  sqrt0  11148  resqrtcl  11173  rersqrtthlem  11174  sqrtsq  11188