Theorem pimltpnf2f 45418

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 2903	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1917	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) < +∞
4		pimltpnf2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6901	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		nfcv 2903	. . . 4 ⊢ Ⅎ𝑥 <
8		nfcv 2903	. . . 4 ⊢ Ⅎ𝑥+∞
9	6, 7, 8	nfbr 5195	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) < +∞
10		fveq2 6891	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq1d 5158	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < +∞ ↔ (𝐹‘𝑦) < +∞))
12	1, 2, 3, 9, 11	cbvrabw 3467	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞}
13		nfv 1917	. . 3 ⊢ Ⅎ𝑦𝜑
14		pimltpnf2f.3	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
15	14	ffvelcdmda 7086	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
16	13, 15	pimltpnf 45410	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} = 𝐴)
17	12, 16	eqtrid 2784	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

Syntax hints:   → wi 4   = wceq 1541  Ⅎwnfc 2883  {crab 3432   class class class wbr 5148  ⟶wf 6539  ‘cfv 6543  ℝcr 11108  +∞cpnf 11244   < clt 11247

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-pnf 11249  df-xr 11251  df-ltxr 11252

This theorem is referenced by:  pimltpnf2  45419  smfpimltxr  45453