Theorem pimltpnf2 44137

Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
pimltpnf2.1	⊢ Ⅎ𝑥𝐹
pimltpnf2.2	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)

Assertion

Ref	Expression
pimltpnf2	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem pimltpnf2

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		pimltpnf2.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5		nfcv 2906	. . . . . 6 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6766	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑥 <
8		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑥+∞
9	6, 7, 8	nfbr 5117	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) < +∞
10		fveq2 6756	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq1d 5080	. . . 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		nfv 1918	. . 3 ⊢ Ⅎ𝑦𝜑
15		pimltpnf2.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
16	15	ffvelrnda 6943	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
17	14, 16	pimltpnf 44130	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} = 𝐴)
18	13, 17	eqtrd 2778	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

Syntax hints:   → wi 4   = wceq 1539  Ⅎwnfc 2886  {crab 3067   class class class wbr 5070  ⟶wf 6414  ‘cfv 6418  ℝcr 10801  +∞cpnf 10937   < clt 10940

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

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  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-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  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-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-pnf 10942  df-xr 10944  df-ltxr 10945

This theorem is referenced by:  smfpimltxr  44170