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Theorem flval 12590
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 (𝑥𝐴𝐴 < (𝑥 + 1))))
Distinct variable group:   𝑥,𝐴

Proof of Theorem flval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 breq2 4655 . . . 4 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
2 breq1 4654 . . . 4 (𝑦 = 𝐴 → (𝑦 < (𝑥 + 1) ↔ 𝐴 < (𝑥 + 1)))
31, 2anbi12d 747 . . 3 (𝑦 = 𝐴 → ((𝑥𝑦𝑦 < (𝑥 + 1)) ↔ (𝑥𝐴𝐴 < (𝑥 + 1))))
43riotabidv 6610 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
5 df-fl 12588 . 2 ⌊ = (𝑦 ∈ ℝ ↦ (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))))
6 riotaex 6612 . 2 (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))) ∈ V
74, 5, 6fvmpt 6280 1 (𝐴 ∈ ℝ → (⌊‘𝐴) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1482  wcel 1989   class class class wbr 4651  cfv 5886  crio 6607  (class class class)co 6647  cr 9932  1c1 9934   + caddc 9936   < clt 10071  cle 10072  cz 11374  cfl 12586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-riota 6608  df-fl 12588
This theorem is referenced by:  flcl  12591  fllelt  12593  flflp1  12603  flbi  12612  dfceil2  12635  ltflcei  33377
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