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 3983 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴) |
3 | ssid 3913 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴) |
5 | pimgtmnf2.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
6 | 5 | ffvelrnda 6893 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
7 | 6 | mnfltd 12699 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦)) |
8 | 7 | ralrimiva 3098 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦)) |
9 | nfcv 2900 | . . . . . . 7 ⊢ Ⅎ𝑥-∞ | |
10 | nfcv 2900 | . . . . . . 7 ⊢ Ⅎ𝑥 < | |
11 | pimgtmnf2.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
12 | nfcv 2900 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
13 | 11, 12 | nffv 6716 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
14 | 9, 10, 13 | nfbr 5090 | . . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦) |
15 | nfv 1922 | . . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥) | |
16 | fveq2 6706 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) | |
17 | 16 | breq2d 5055 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (-∞ < (𝐹‘𝑦) ↔ -∞ < (𝐹‘𝑥))) |
18 | 14, 15, 17 | cbvralw 3342 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦) ↔ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)) |
19 | 8, 18 | sylib 221 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)) |
20 | 4, 19 | jca 515 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))) |
21 | nfcv 2900 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
22 | 21, 21 | ssrabf 42289 | . . 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 3908 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Ⅎwnfc 2880 ∀wral 3054 {crab 3058 ⊆ wss 3857 class class class wbr 5043 ⟶wf 6365 ‘cfv 6369 ℝcr 10711 -∞cmnf 10848 < clt 10850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 |
This theorem is referenced by: pimgtmnf 43885 |
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