Theorem pimgtpnf2f 45411

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 1917	. . 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 6901	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
9	4, 5, 8	nfbr 5195	. . 3 ⊢ Ⅎ𝑥+∞ < (𝐹‘𝑦)
10		fveq2 6891	. . . 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 3467	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)}
13		pimgtpnf2f.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
14	13	ffvelcdmda 7086	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
15	14	rexrd 11263	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*)
16	15	pnfged 44174	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ≤ +∞)
17		pnfxr 11267	. . . . . . 7 ⊢ +∞ ∈ ℝ*
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → +∞ ∈ ℝ*)
19	15, 18	xrlenltd 11279	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹‘𝑦)))
20	16, 19	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ +∞ < (𝐹‘𝑦))
21	20	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
22		rabeq0 4384	. . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
23	21, 22	sylibr 233	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅)
24	12, 23	eqtrid 2784	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Ⅎwnfc 2883  ∀wral 3061  {crab 3432  ∅c0 4322   class class class wbr 5148  ⟶wf 6539  ‘cfv 6543  ℝcr 11108  +∞cpnf 11244  ℝ^*cxr 11246   < clt 11247   ≤ cle 11248

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253

This theorem is referenced by:  pimgtpnf2  45412  smfpimgtxr  45486