Theorem pimltpnf2f 46834

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 1915	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) < +∞
4		pimltpnf2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2895	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6838	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		nfcv 2895	. . . 4 ⊢ Ⅎ𝑥 <
8		nfcv 2895	. . . 4 ⊢ Ⅎ𝑥+∞
9	6, 7, 8	nfbr 5140	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) < +∞
10		fveq2 6828	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq1d 5103	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < +∞ ↔ (𝐹‘𝑦) < +∞))
12	1, 2, 3, 9, 11	cbvrabw 3431	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞}
13		nfv 1915	. . 3 ⊢ Ⅎ𝑦𝜑
14		pimltpnf2f.3	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
15	14	ffvelcdmda 7023	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
16	13, 15	pimltpnf 46826	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} = 𝐴)
17	12, 16	eqtrid 2780	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

Syntax hints:   → wi 4   = wceq 1541  Ⅎwnfc 2880  {crab 3396   class class class wbr 5093  ⟶wf 6482  ‘cfv 6486  ℝcr 11012  +∞cpnf 11150   < clt 11153

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-pnf 11155  df-xr 11157  df-ltxr 11158

This theorem is referenced by:  pimltpnf2  46835  smfpimltxr  46869