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Theorem pimgtmnf 43723
 Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
pimgtmnf.1 𝑥𝜑
pimgtmnf.2 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
Assertion
Ref Expression
pimgtmnf (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem pimgtmnf
StepHypRef Expression
1 pimgtmnf.1 . . 3 𝑥𝜑
2 eqidd 2759 . . . . . 6 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
3 pimgtmnf.2 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
42, 3fvmpt2d 6772 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
54eqcomd 2764 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = ((𝑥𝐴𝐵)‘𝑥))
65breq2d 5044 . . 3 ((𝜑𝑥𝐴) → (-∞ < 𝐵 ↔ -∞ < ((𝑥𝐴𝐵)‘𝑥)))
71, 6rabbida 3386 . 2 (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = {𝑥𝐴 ∣ -∞ < ((𝑥𝐴𝐵)‘𝑥)})
8 nfmpt1 5130 . . 3 𝑥(𝑥𝐴𝐵)
9 eqid 2758 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
101, 3, 9fmptdf 6872 . . 3 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
118, 10pimgtmnf2 43715 . 2 (𝜑 → {𝑥𝐴 ∣ -∞ < ((𝑥𝐴𝐵)‘𝑥)} = 𝐴)
127, 11eqtrd 2793 1 (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  {crab 3074   class class class wbr 5032   ↦ cmpt 5112  ‘cfv 6335  ℝcr 10574  -∞cmnf 10711   < clt 10713 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718 This theorem is referenced by:  smfpimgtxr  43779
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