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.

Step	Hyp	Ref	Expression
1		eqeq2 2104	. . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥↑2) = 𝑦 ↔ (𝑥↑2) = 𝐴))
2	1	anbi1d 454	. . 3 ⊢ (𝑦 = 𝐴 → (((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥) ↔ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
3	2	riotabidv 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)
7	5, 6	ax-mp 7	. 2 ⊢ (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) ∈ V
8	3, 4, 7	fvmpt 5416	1 ⊢ (𝐴 ∈ ℝ → (√‘𝐴) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))

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  ℝcr 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