Theorem sqrtrval 11022

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 2197	. . . 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 5846	. 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥)) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))
4		df-rsqrt 11020	. 2 ⊢ √ = (𝑦 ∈ ℝ ↦ (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥)))
5		reex 7958	. . 3 ⊢ ℝ ∈ V
6		riotaexg 5848	. . 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 5606	1 ⊢ (𝐴 ∈ ℝ → (√‘𝐴) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1363   ∈ wcel 2158  Vcvv 2749   class class class wbr 4015  ‘cfv 5228  ℩crio 5843  (class class class)co 5888  ℝcr 7823  0cc0 7824   ≤ cle 8006  2c2 8983  ↑cexp 10532  √csqrt 11018

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7915  ax-resscn 7916

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-rsqrt 11020

This theorem is referenced by:  sqrt0  11026  resqrtcl  11051  rersqrtthlem  11052  sqrtsq  11066