Theorem pimgtpnf2f 47520

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 2924	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1947	. . 3 ⊢ Ⅎ𝑦+∞ < (𝐹‘𝑥)
4		nfcv 2924	. . . 4 ⊢ Ⅎ𝑥+∞
5		nfcv 2924	. . . 4 ⊢ Ⅎ𝑥 <
6		pimgtpnf2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
7		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑥𝑦
8	6, 7	nffv 6892	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
9	4, 5, 8	nfbr 5156	. . 3 ⊢ Ⅎ𝑥+∞ < (𝐹‘𝑦)
10		fveq2 6882	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq2d 5119	. . 3 ⊢ (𝑥 = 𝑦 → (+∞ < (𝐹‘𝑥) ↔ +∞ < (𝐹‘𝑦)))
12	1, 2, 3, 9, 11	cbvrabw 3449	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)}
13		pimgtpnf2f.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
14	13	ffvelcdmda 7080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
15	14	rexrd 11286	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*)
16	15	pnfged 13184	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ≤ +∞)
17		pnfxr 11290	. . . . . . 7 ⊢ +∞ ∈ ℝ*
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → +∞ ∈ ℝ*)
19	15, 18	xrlenltd 11302	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹‘𝑦)))
20	16, 19	mpbid 235	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ +∞ < (𝐹‘𝑦))
21	20	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
22		rabeq0 4341	. . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
23	21, 22	sylibr 237	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅)
24	12, 23	eqtrid 2809	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2909  ∀wral 3078  {crab 3414  ∅c0 4282   class class class wbr 5107  ⟶wf 6533  ‘cfv 6537  ℝcr 11126  +∞cpnf 11267  ℝ^*cxr 11269   < clt 11270   ≤ cle 11271

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276

This theorem is used by:  pimgtpnf2  47521  smfpimgtxr  47595