Theorem pimltpnf2 43348

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 2955	. . . 4 ⊢ Ⅎ𝑥𝐴
2		nfcv 2955	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfv 1915	. . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑥) < +∞
4		pimltpnf2.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6655	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑥 <
8		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑥+∞
9	6, 7, 8	nfbr 5077	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) < +∞
10		fveq2 6645	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq1d 5040	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < +∞ ↔ (𝐹‘𝑦) < +∞))
12	1, 2, 3, 9, 11	cbvrabw 3437	. . 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 1915	. . 3 ⊢ Ⅎ𝑦𝜑
15		pimltpnf2.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
16	15	ffvelrnda 6828	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
17	14, 16	pimltpnf 43341	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} = 𝐴)
18	13, 17	eqtrd 2833	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

Syntax hints:   → wi 4   = wceq 1538  Ⅎwnfc 2936  {crab 3110   class class class wbr 5030  ⟶wf 6320  ‘cfv 6324  ℝcr 10525  +∞cpnf 10661   < clt 10664

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-pnf 10666  df-xr 10668  df-ltxr 10669

This theorem is referenced by:  smfpimltxr  43381