![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > pimgtmnf2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
pimgtmnf2.1 | ⊢ Ⅎ𝑥𝐹 |
pimgtmnf2.2 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
Ref | Expression |
---|---|
pimgtmnf2 | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4090 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴) |
3 | ssid 4018 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴) |
5 | pimgtmnf2.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
6 | 5 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
7 | 6 | mnfltd 13164 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦)) |
8 | 7 | ralrimiva 3144 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦)) |
9 | nfcv 2903 | . . . . . . 7 ⊢ Ⅎ𝑥-∞ | |
10 | nfcv 2903 | . . . . . . 7 ⊢ Ⅎ𝑥 < | |
11 | pimgtmnf2.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
12 | nfcv 2903 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
13 | 11, 12 | nffv 6917 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
14 | 9, 10, 13 | nfbr 5195 | . . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦) |
15 | nfv 1912 | . . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥) | |
16 | fveq2 6907 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) | |
17 | 16 | breq2d 5160 | . . . . . 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 218 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)) |
20 | 4, 19 | jca 511 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))) |
21 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
22 | 21, 21 | ssrabf 45054 | . . 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 4013 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Ⅎwnfc 2888 ∀wral 3059 {crab 3433 ⊆ wss 3963 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 ℝcr 11152 -∞cmnf 11291 < clt 11293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |