Theorem pimgtpnf2f 46661

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 2903	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1912	. . 3 ⊢ Ⅎ𝑦+∞ < (𝐹‘𝑥)
4		nfcv 2903	. . . 4 ⊢ Ⅎ𝑥+∞
5		nfcv 2903	. . . 4 ⊢ Ⅎ𝑥 <
6		pimgtpnf2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
7		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥𝑦
8	6, 7	nffv 6917	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
9	4, 5, 8	nfbr 5195	. . 3 ⊢ Ⅎ𝑥+∞ < (𝐹‘𝑦)
10		fveq2 6907	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq2d 5160	. . 3 ⊢ (𝑥 = 𝑦 → (+∞ < (𝐹‘𝑥) ↔ +∞ < (𝐹‘𝑦)))
12	1, 2, 3, 9, 11	cbvrabw 3471	. 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 11309	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*)
16	15	pnfged 45424	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ≤ +∞)
17		pnfxr 11313	. . . . . . 7 ⊢ +∞ ∈ ℝ*
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → +∞ ∈ ℝ*)
19	15, 18	xrlenltd 11325	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹‘𝑦)))
20	16, 19	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ +∞ < (𝐹‘𝑦))
21	20	ralrimiva 3144	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
22		rabeq0 4394	. . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
23	21, 22	sylibr 234	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅)
24	12, 23	eqtrid 2787	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Ⅎwnfc 2888  ∀wral 3059  {crab 3433  ∅c0 4339   class class class wbr 5148  ⟶wf 6559  ‘cfv 6563  ℝcr 11152  +∞cpnf 11290  ℝ^*cxr 11292   < clt 11293   ≤ cle 11294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299

This theorem is referenced by:  pimgtpnf2  46662  smfpimgtxr  46736