Theorem pimgtpnf2 42979

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 2977	. . . 4 ⊢ Ⅎ𝑥𝐴
2		nfcv 2977	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfv 1911	. . . 4 ⊢ Ⅎ𝑦+∞ < (𝐹‘𝑥)
4		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑥+∞
5		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑥 <
6		pimgtpnf2.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
7		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑥𝑦
8	6, 7	nffv 6674	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦)
9	4, 5, 8	nfbr 5105	. . . 4 ⊢ Ⅎ𝑥+∞ < (𝐹‘𝑦)
10		fveq2 6664	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq2d 5070	. . . 4 ⊢ (𝑥 = 𝑦 → (+∞ < (𝐹‘𝑥) ↔ +∞ < (𝐹‘𝑦)))
12	1, 2, 3, 9, 11	cbvrabw 3489	. . 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 6845	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
16	15	rexrd 10685	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ*)
17		pnfxr 10689	. . . . . . 7 ⊢ +∞ ∈ ℝ*
18	17	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → +∞ ∈ ℝ*)
19	15	ltpnfd 12510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) < +∞)
20	16, 18, 19	xrltled 12537	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ≤ +∞)
21	16, 18	xrlenltd 10701	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) ≤ +∞ ↔ ¬ +∞ < (𝐹‘𝑦)))
22	20, 21	mpbid 234	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ +∞ < (𝐹‘𝑦))
23	22	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
24		rabeq0 4337	. . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅ ↔ ∀𝑦 ∈ 𝐴 ¬ +∞ < (𝐹‘𝑦))
25	23, 24	sylibr 236	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑦)} = ∅)
26	13, 25	eqtrd 2856	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Ⅎwnfc 2961  ∀wral 3138  {crab 3142  ∅c0 4290   class class class wbr 5058  ⟶wf 6345  ‘cfv 6349  ℝcr 10530  +∞cpnf 10666  ℝ^*cxr 10668   < clt 10669   ≤ cle 10670

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-pre-lttri 10605  ax-pre-lttrn 10606

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-po 5468  df-so 5469  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675

This theorem is referenced by:  smfpimgtxr  43050