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Theorem pimgtmnf 41449
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 2772 . . . . . 6 (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐴𝐵))
3 pimgtmnf.2 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
42, 3fvmpt2d 6437 . . . . 5 ((𝜑𝑥𝐴) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
54eqcomd 2777 . . . 4 ((𝜑𝑥𝐴) → 𝐵 = ((𝑥𝐴𝐵)‘𝑥))
65breq2d 4799 . . 3 ((𝜑𝑥𝐴) → (-∞ < 𝐵 ↔ -∞ < ((𝑥𝐴𝐵)‘𝑥)))
71, 6rabbida 39795 . 2 (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = {𝑥𝐴 ∣ -∞ < ((𝑥𝐴𝐵)‘𝑥)})
8 nfmpt1 4882 . . 3 𝑥(𝑥𝐴𝐵)
9 eqid 2771 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
101, 3, 9fmptdf 6532 . . 3 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
118, 10pimgtmnf2 41441 . 2 (𝜑 → {𝑥𝐴 ∣ -∞ < ((𝑥𝐴𝐵)‘𝑥)} = 𝐴)
127, 11eqtrd 2805 1 (𝜑 → {𝑥𝐴 ∣ -∞ < 𝐵} = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wnf 1856  wcel 2145  {crab 3065   class class class wbr 4787  cmpt 4864  cfv 6030  cr 10141  -∞cmnf 10278   < clt 10280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285
This theorem is referenced by:  smfpimgtxr  41505
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