Theorem pimltpnf2 43011

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 2977	. . . 4 ⊢ Ⅎ𝑥𝐴
2		nfcv 2977	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfv 1915	. . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑥) < +∞
4		pimltpnf2.1	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6680	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑥 <
8		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑥+∞
9	6, 7, 8	nfbr 5113	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) < +∞
10		fveq2 6670	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq1d 5076	. . . 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		nfv 1915	. . 3 ⊢ Ⅎ𝑦𝜑
15		pimltpnf2.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
16	15	ffvelrnda 6851	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
17	14, 16	pimltpnf 43004	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} = 𝐴)
18	13, 17	eqtrd 2856	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

Syntax hints:   → wi 4   = wceq 1537  Ⅎwnfc 2961  {crab 3142   class class class wbr 5066  ⟶wf 6351  ‘cfv 6355  ℝcr 10536  +∞cpnf 10672   < clt 10675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-pnf 10677  df-xr 10679  df-ltxr 10680

This theorem is referenced by:  smfpimltxr  43044