Theorem flval 13801

Description: Value of the floor (greatest integer) function. The floor of 𝐴 is the (unique) integer less than or equal to 𝐴 whose successor is strictly greater than 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by Mario Carneiro, 2-Nov-2013.)

Assertion

Ref	Expression
flval	⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))

Distinct variable group:   𝑥,𝐴

Proof of Theorem flval

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5103	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ 𝐴))
2		breq1 5102	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 < (𝑥 + 1) ↔ 𝐴 < (𝑥 + 1)))
3	1, 2	anbi12d 641	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1)) ↔ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
4	3	riotabidv 7351	. 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))
5		df-fl 13799	. 2 ⊢ ⌊ = (𝑦 ∈ ℝ ↦ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝑦 ∧ 𝑦 < (𝑥 + 1))))
6		riotaex 7353	. 2 ⊢ (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))) ∈ V
7	4, 5, 6	fvmpt 6971	1 ⊢ (𝐴 ∈ ℝ → (⌊‘𝐴) = (℩𝑥 ∈ ℤ (𝑥 ≤ 𝐴 ∧ 𝐴 < (𝑥 + 1))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141   class class class wbr 5099  ‘cfv 6517  ℩crio 7348  (class class class)co 7392  ℝcr 11069  1c1 11071   + caddc 11073   < clt 11213   ≤ cle 11214  ℤcz 12565  ⌊cfl 13797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-riota 7349  df-fl 13799

This theorem is referenced by:  flcl  13802  fllelt  13804  flflp1  13814  flbi  13823  dfceil2  13846  ltflcei  38071