Theorem pimgtmnf2 46811

Description: Given a real-valued function, the preimage of an open interval, unbounded above, with lower bound -∞, is the whole domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
pimgtmnf2.1	⊢ Ⅎ𝑥𝐹
pimgtmnf2.2	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)

Assertion

Ref	Expression
pimgtmnf2	⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem pimgtmnf2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 4027	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴)
3		ssid 3952	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
5		pimgtmnf2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
6	5	ffvelcdmda 7017	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
7	6	mnfltd 13023	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦))
8	7	ralrimiva 3124	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦))
9		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑥-∞
10		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑥 <
11		pimgtmnf2.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
12		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
13	11, 12	nffv 6832	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦)
14	9, 10, 13	nfbr 5136	. . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦)
15		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥)
16		fveq2 6822	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
17	16	breq2d 5101	. . . . . 6 ⊢ (𝑦 = 𝑥 → (-∞ < (𝐹‘𝑦) ↔ -∞ < (𝐹‘𝑥)))
18	14, 15, 17	cbvralw 3274	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦) ↔ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))
19	8, 18	sylib 218	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))
20	4, 19	jca 511	. . 3 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)))
21		nfcv 2894	. . . 4 ⊢ Ⅎ𝑥𝐴
22	21, 21	ssrabf 45210	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)))
23	20, 22	sylibr 234	. 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)})
24	2, 23	eqssd 3947	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Ⅎwnfc 2879  ∀wral 3047  {crab 3395   ⊆ wss 3897   class class class wbr 5089  ⟶wf 6477  ‘cfv 6481  ℝcr 11005  -∞cmnf 11144   < clt 11146

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151

This theorem is referenced by: (None)