Theorem pimgtpnf2 47716

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 2922	. 2 ⊢ Ⅎ𝑥𝐴
3		pimgtpnf2.2	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
4	1, 2, 3	pimgtpnf2f 47715	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  Ⅎwnfc 2907  {crab 3412  ∅c0 4278   class class class wbr 5102  ⟶wf 6534  ‘cfv 6538  ℝcr 11199  +∞cpnf 11340   < clt 11343

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349

This theorem is used by: (None)