Theorem pimltpnf2f 46444

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, 15-Dec-2024.)

Hypotheses

Ref	Expression
pimltpnf2f.1	⊢ Ⅎ𝑥𝐹
pimltpnf2f.2	⊢ Ⅎ𝑥𝐴
pimltpnf2f.3	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)

Assertion

Ref	Expression
pimltpnf2f	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

Proof of Theorem pimltpnf2f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pimltpnf2f.2	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfcv 2895	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1910	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) < +∞
4		pimltpnf2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2895	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6913	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		nfcv 2895	. . . 4 ⊢ Ⅎ𝑥 <
8		nfcv 2895	. . . 4 ⊢ Ⅎ𝑥+∞
9	6, 7, 8	nfbr 5201	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) < +∞
10		fveq2 6903	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq1d 5164	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < +∞ ↔ (𝐹‘𝑦) < +∞))
12	1, 2, 3, 9, 11	cbvrabw 3465	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞}
13		nfv 1910	. . 3 ⊢ Ⅎ𝑦𝜑
14		pimltpnf2f.3	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
15	14	ffvelcdmda 7100	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
16	13, 15	pimltpnf 46436	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} = 𝐴)
17	12, 16	eqtrid 2781	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

Syntax hints:   → wi 4   = wceq 1534  Ⅎwnfc 2879  {crab 3428   class class class wbr 5154  ⟶wf 6552  ‘cfv 6556  ℝcr 11158  +∞cpnf 11296   < clt 11299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-sep 5305  ax-nul 5312  ax-pow 5371  ax-pr 5435  ax-un 7748  ax-cnex 11215

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-ral 3055  df-rex 3064  df-rab 3429  df-v 3474  df-dif 3961  df-un 3963  df-ss 3975  df-nul 4334  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4917  df-br 5155  df-opab 5217  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6508  df-fun 6558  df-fn 6559  df-f 6560  df-fv 6564  df-pnf 11301  df-xr 11303  df-ltxr 11304

This theorem is referenced by:  pimltpnf2  46445  smfpimltxr  46479