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Theorem flval 10075
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 3940 . . . 4 (𝑦 = 𝐴 → (𝑥𝑦𝑥𝐴))
2 breq1 3939 . . . 4 (𝑦 = 𝐴 → (𝑦 < (𝑥 + 1) ↔ 𝐴 < (𝑥 + 1)))
31, 2anbi12d 465 . . 3 (𝑦 = 𝐴 → ((𝑥𝑦𝑦 < (𝑥 + 1)) ↔ (𝑥𝐴𝐴 < (𝑥 + 1))))
43riotabidv 5739 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
5 df-fl 10073 . 2 ⌊ = (𝑦 ∈ ℝ ↦ (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))))
6 zex 9086 . . 3 ℤ ∈ V
7 riotaexg 5741 . . 3 (ℤ ∈ V → (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))) ∈ V)
86, 7ax-mp 5 . 2 (𝑥 ∈ ℤ (𝑥𝑦𝑦 < (𝑥 + 1))) ∈ V
94, 5, 8fvmpt3i 5508 1 (𝐴 ∈ ℝ → (⌊‘𝐴) = (𝑥 ∈ ℤ (𝑥𝐴𝐴 < (𝑥 + 1))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  Vcvv 2689   class class class wbr 3936  cfv 5130  crio 5736  (class class class)co 5781  cr 7642  1c1 7644   + caddc 7646   < clt 7823  cle 7824  cz 9077  cfl 10071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-cnex 7734  ax-resscn 7735
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fv 5138  df-riota 5737  df-ov 5784  df-neg 7959  df-z 9078  df-fl 10073
This theorem is referenced by:  flqcl  10076  flapcl  10078  flqlelt  10079  flqbi  10093
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