Theorem pimgtpnf2 44131

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, 26-Jun-2021.)

Hypotheses

Ref	Expression
pimgtpnf2.1	⊢ Ⅎ𝑥𝐹
pimgtpnf2.2	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)

Assertion

Ref	Expression
pimgtpnf2	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem pimgtpnf2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2906	. . . 4 ⊢ Ⅎ𝑥𝐴
2		nfcv 2906	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfv 1918	. . . 4 ⊢ Ⅎ𝑦+∞ < (𝐹‘𝑥)
4		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑥+∞
5		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑥 <
6		pimgtpnf2.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
7		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑥𝑦
8	6, 7	nffv 6766	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦)
9	4, 5, 8	nfbr 5117	. . . 4 ⊢ Ⅎ𝑥+∞ < (𝐹‘𝑦)
10		fveq2 6756	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq2d 5082	. . . 4 ⊢ (𝑥 = 𝑦 → (+∞ < (𝐹‘𝑥) ↔ +∞ < (𝐹‘𝑦)))
12	1, 2, 3, 9, 11	cbvrabw 3414	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)}
13	12	a1i 11	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)})
14		pimgtpnf2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
15	14	ffvelrnda 6943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
16	15	rexrd 10956	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*)
17		pnfxr 10960	. . . . . . 7 ⊢ +∞ ∈ ℝ*
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → +∞ ∈ ℝ*)
19	15	ltpnfd 12786	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) < +∞)
20	16, 18, 19	xrltled 12813	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ≤ +∞)
21	16, 18	xrlenltd 10972	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹‘𝑦)))
22	20, 21	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ +∞ < (𝐹‘𝑦))
23	22	ralrimiva 3107	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
24		rabeq0 4315	. . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
25	23, 24	sylibr 233	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅)
26	13, 25	eqtrd 2778	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Ⅎwnfc 2886  ∀wral 3063  {crab 3067  ∅c0 4253   class class class wbr 5070  ⟶wf 6414  ‘cfv 6418  ℝcr 10801  +∞cpnf 10937  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-pre-lttri 10876  ax-pre-lttrn 10877

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946

This theorem is referenced by:  smfpimgtxr  44202