Theorem pimgtmnf2 47456

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 4033	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴)
3		ssid 3958	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
5		pimgtmnf2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
6	5	ffvelcdmda 7079	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
7	6	mnfltd 13155	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦))
8	7	ralrimiva 3156	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦))
9		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑥-∞
10		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑥 <
11		pimgtmnf2.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
12		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
13	11, 12	nffv 6891	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦)
14	9, 10, 13	nfbr 5157	. . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦)
15		nfv 1943	. . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥)
16		fveq2 6881	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
17	16	breq2d 5120	. . . . . 6 ⊢ (𝑦 = 𝑥 → (-∞ < (𝐹‘𝑦) ↔ -∞ < (𝐹‘𝑥)))
18	14, 15, 17	cbvralw 3306	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦) ↔ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))
19	8, 18	sylib 221	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))
20	4, 19	jca 520	. . 3 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)))
21		nfcv 2924	. . . 4 ⊢ Ⅎ𝑥𝐴
22	21, 21	ssrabf 45860	. . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)))
23	20, 22	sylibr 237	. 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)})
24	2, 23	eqssd 3953	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1569   ∈ wcel 2142  Ⅎwnfc 2909  ∀wral 3078  {crab 3415   ⊆ wss 3904   class class class wbr 5108  ⟶wf 6532  ‘cfv 6536  ℝcr 11105  -∞cmnf 11247   < clt 11249

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254

This theorem is used by: (None)