Theorem pimgtmnf2 47724

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 4005	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴)
3		ssid 3930	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
4	3	a1i 11	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
5		pimgtmnf2.2	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
6	5	ffvelcdmda 7084	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ)
7	6	mnfltd 13253	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦))
8	7	ralrimiva 3154	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦))
9		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑥-∞
10		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑥 <
11		pimgtmnf2.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
12		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
13	11, 12	nffv 6895	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦)
14	9, 10, 13	nfbr 5151	. . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦)
15		nfv 1946	. . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥)
16		fveq2 6885	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥))
17	16	breq2d 5114	. . . . . 6 ⊢ (𝑦 = 𝑥 → (-∞ < (𝐹‘𝑦) ↔ -∞ < (𝐹‘𝑥)))
18	14, 15, 17	cbvralw 3304	. . . . 5 ⊢ (∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦) ↔ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))
19	8, 18	sylib 221	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))
20	4, 19	jca 521	. . 3 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)))
21		nfcv 2922	. . . 4 ⊢ Ⅎ𝑥𝐴
22	21, 21	ssrabf 46129	. . 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 3925	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Ⅎwnfc 2907  ∀wral 3076  {crab 3412   ⊆ wss 3895   class class class wbr 5102  ⟶wf 6534  ‘cfv 6538  ℝcr 11199  -∞cmnf 11341   < clt 11343

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348

This theorem is used by: (None)