Theorem pimgtpnf2 47279

Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound +∞, is the empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.)

Hypotheses

Ref	Expression
pimgtpnf2.1	⊢ Ⅎ𝑥𝐹
pimgtpnf2.2	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)

Assertion

Ref	Expression
pimgtpnf2	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem pimgtpnf2

Step	Hyp	Ref	Expression
1		pimgtpnf2.1	. 2 ⊢ Ⅎ𝑥𝐹
2		nfcv 2927	. 2 ⊢ Ⅎ𝑥𝐴
3		pimgtpnf2.2	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
4	1, 2, 3	pimgtpnf2f 47278	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

Syntax hints:   → wi 4   = wceq 1563  Ⅎwnfc 2912  {crab 3417  ∅c0 4288   class class class wbr 5104  ⟶wf 6521  ‘cfv 6525  ℝcr 11087  +∞cpnf 11228   < clt 11231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-cnex 11144  ax-resscn 11145

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237

This theorem is referenced by: (None)