Theorem pimgtpnf2f 46986

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 2897	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1916	. . 3 ⊢ Ⅎ𝑦+∞ < (𝐹‘𝑥)
4		nfcv 2897	. . . 4 ⊢ Ⅎ𝑥+∞
5		nfcv 2897	. . . 4 ⊢ Ⅎ𝑥 <
6		pimgtpnf2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
7		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑥𝑦
8	6, 7	nffv 6843	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
9	4, 5, 8	nfbr 5144	. . 3 ⊢ Ⅎ𝑥+∞ < (𝐹‘𝑦)
10		fveq2 6833	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq2d 5109	. . 3 ⊢ (𝑥 = 𝑦 → (+∞ < (𝐹‘𝑥) ↔ +∞ < (𝐹‘𝑦)))
12	1, 2, 3, 9, 11	cbvrabw 3433	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)}
13		pimgtpnf2f.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
14	13	ffvelcdmda 7029	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
15	14	rexrd 11184	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*)
16	15	pnfged 13047	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ≤ +∞)
17		pnfxr 11188	. . . . . . 7 ⊢ +∞ ∈ ℝ*
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → +∞ ∈ ℝ*)
19	15, 18	xrlenltd 11200	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹‘𝑦)))
20	16, 19	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ +∞ < (𝐹‘𝑦))
21	20	ralrimiva 3127	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
22		rabeq0 4339	. . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
23	21, 22	sylibr 234	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅)
24	12, 23	eqtrid 2782	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2882  ∀wral 3050  {crab 3398  ∅c0 4284   class class class wbr 5097  ⟶wf 6487  ‘cfv 6491  ℝcr 11027  +∞cpnf 11165  ℝ^*cxr 11167   < clt 11168   ≤ cle 11169

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-fv 6499  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174

This theorem is referenced by:  pimgtpnf2  46987  smfpimgtxr  47061