Mathbox for Glauco Siliprandi |
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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 4055 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴) |
3 | ssid 3988 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴) |
5 | pimgtmnf2.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
6 | 5 | ffvelrnda 6850 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
7 | 6 | mnfltd 12518 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦)) |
8 | 7 | ralrimiva 3182 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦)) |
9 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑥-∞ | |
10 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑥 < | |
11 | pimgtmnf2.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
12 | nfcv 2977 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
13 | 11, 12 | nffv 6679 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
14 | 9, 10, 13 | nfbr 5112 | . . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦) |
15 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥) | |
16 | fveq2 6669 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) | |
17 | 16 | breq2d 5077 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (-∞ < (𝐹‘𝑦) ↔ -∞ < (𝐹‘𝑥))) |
18 | 14, 15, 17 | cbvralw 3441 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦) ↔ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)) |
19 | 8, 18 | sylib 220 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)) |
20 | 4, 19 | jca 514 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))) |
21 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
22 | 21, 21 | ssrabf 41379 | . . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))) |
23 | 20, 22 | sylibr 236 | . 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)}) |
24 | 2, 23 | eqssd 3983 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Ⅎwnfc 2961 ∀wral 3138 {crab 3142 ⊆ wss 3935 class class class wbr 5065 ⟶wf 6350 ‘cfv 6354 ℝcr 10535 -∞cmnf 10672 < clt 10674 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 |
This theorem is referenced by: pimgtmnf 42999 |
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