Theorem pimltpnf2f 47937

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 2926	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1949	. . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) < +∞
4		pimltpnf2f.1	. . . . 5 ⊢ Ⅎ𝑥𝐹
5		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥𝑦
6	4, 5	nffv 6921	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦)
7		nfcv 2926	. . . 4 ⊢ Ⅎ𝑥 <
8		nfcv 2926	. . . 4 ⊢ Ⅎ𝑥+∞
9	6, 7, 8	nfbr 5156	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) < +∞
10		fveq2 6911	. . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
11	10	breq1d 5116	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < +∞ ↔ (𝐹‘𝑦) < +∞))
12	1, 2, 3, 9, 11	cbvrabw 3450	. 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞}
13		nfv 1949	. . 3 ⊢ Ⅎ𝑦𝜑
14		pimltpnf2f.3	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
15	14	ffvelcdmda 7111	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
16	13, 15	pimltpnf 47929	. 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} = 𝐴)
17	12, 16	eqtrid 2810	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Ⅎwnfc 2911  {crab 3416   class class class wbr 5106  ⟶wf 6557  ‘cfv 6561  ℝcr 11229  +∞cpnf 11370   < clt 11373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-pnf 11375  df-xr 11377  df-ltxr 11378

This theorem is used by:  pimltpnf2  47938  smfpimltxr  47972