Theorem pimgtpnf2 46987

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 2897	. 2 ⊢ Ⅎ𝑥𝐴
3		pimgtpnf2.2	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
4	1, 2, 3	pimgtpnf2f 46986	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

Syntax hints:   → wi 4   = wceq 1542  Ⅎwnfc 2882  {crab 3398  ∅c0 4284   class class class wbr 5097  ⟶wf 6487  ‘cfv 6491  ℝcr 11027  +∞cpnf 11165   < clt 11168

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-fv 6499  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174

This theorem is referenced by: (None)