Theorem pimgtmnf2 47142

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 4020	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴)
3		ssid 3944	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
5		pimgtmnf2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
6	5	ffvelcdmda 7036	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
7	6	mnfltd 13075	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦))
8	7	ralrimiva 3129	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦))
9		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑥-∞
10		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑥 <
11		pimgtmnf2.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
12		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
13	11, 12	nffv 6850	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦)
14	9, 10, 13	nfbr 5132	. . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦)
15		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥)
16		fveq2 6840	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
17	16	breq2d 5097	. . . . . 6 ⊢ (𝑦 = 𝑥 → (-∞ < (𝐹‘𝑦) ↔ -∞ < (𝐹‘𝑥)))
18	14, 15, 17	cbvralw 3279	. . . . 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 2898	. . . 4 ⊢ Ⅎ𝑥𝐴
22	21, 21	ssrabf 45544	. . 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 3939	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2883  ∀wral 3051  {crab 3389   ⊆ wss 3889   class class class wbr 5085  ⟶wf 6494  ‘cfv 6498  ℝcr 11037  -∞cmnf 11177   < clt 11179

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184

This theorem is referenced by: (None)