Theorem pimgtmnf2 47355

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 4040	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴)
3		ssid 3965	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
5		pimgtmnf2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
6	5	ffvelcdmda 7080	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
7	6	mnfltd 13149	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦))
8	7	ralrimiva 3163	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦))
9		nfcv 2931	. . . . . . 7 ⊢ Ⅎ𝑥-∞
10		nfcv 2931	. . . . . . 7 ⊢ Ⅎ𝑥 <
11		pimgtmnf2.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
12		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
13	11, 12	nffv 6892	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦)
14	9, 10, 13	nfbr 5160	. . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦)
15		nfv 1941	. . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥)
16		fveq2 6882	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
17	16	breq2d 5123	. . . . . 6 ⊢ (𝑦 = 𝑥 → (-∞ < (𝐹‘𝑦) ↔ -∞ < (𝐹‘𝑥)))
18	14, 15, 17	cbvralw 3313	. . . . 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 2931	. . . 4 ⊢ Ⅎ𝑥𝐴
22	21, 21	ssrabf 45759	. . 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 3960	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Ⅎwnfc 2916  ∀wral 3085  {crab 3422   ⊆ wss 3911   class class class wbr 5111  ⟶wf 6533  ‘cfv 6537  ℝcr 11099  -∞cmnf 11241   < clt 11243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248

This theorem is referenced by: (None)