Theorem pimgtpnf2f 46720

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, 15-Dec-2021.)

Hypotheses

Ref	Expression
pimgtpnf2f.1	⊢ Ⅎ𝑥𝐹
pimgtpnf2f.2	⊢ Ⅎ𝑥𝐴
pimgtpnf2f.3	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)

Assertion

Ref	Expression
pimgtpnf2f	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

Proof of Theorem pimgtpnf2f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pimgtpnf2f.2	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfcv 2905	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1914	. . 3 ⊢ Ⅎ𝑦+∞ < (𝐹‘𝑥)
4		nfcv 2905	. . . 4 ⊢ Ⅎ𝑥+∞
5		nfcv 2905	. . . 4 ⊢ Ⅎ𝑥 <
6		pimgtpnf2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
7		nfcv 2905	. . . . 5 ⊢ Ⅎ𝑥𝑦
8	6, 7	nffv 6916	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
9	4, 5, 8	nfbr 5190	. . 3 ⊢ Ⅎ𝑥+∞ < (𝐹‘𝑦)
10		fveq2 6906	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq2d 5155	. . 3 ⊢ (𝑥 = 𝑦 → (+∞ < (𝐹‘𝑥) ↔ +∞ < (𝐹‘𝑦)))
12	1, 2, 3, 9, 11	cbvrabw 3473	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)}
13		pimgtpnf2f.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
14	13	ffvelcdmda 7104	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
15	14	rexrd 11311	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*)
16	15	pnfged 13173	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ≤ +∞)
17		pnfxr 11315	. . . . . . 7 ⊢ +∞ ∈ ℝ*
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → +∞ ∈ ℝ*)
19	15, 18	xrlenltd 11327	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹‘𝑦)))
20	16, 19	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ +∞ < (𝐹‘𝑦))
21	20	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
22		rabeq0 4388	. . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
23	21, 22	sylibr 234	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅)
24	12, 23	eqtrid 2789	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2890  ∀wral 3061  {crab 3436  ∅c0 4333   class class class wbr 5143  ⟶wf 6557  ‘cfv 6561  ℝcr 11154  +∞cpnf 11292  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301

This theorem is referenced by:  pimgtpnf2  46721  smfpimgtxr  46795