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 3650 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴) |
3 | ssid 3587 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴) |
5 | pimgtmnf2.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
6 | 5 | ffvelrnda 6267 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
7 | 6 | mnfltd 11834 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦)) |
8 | 7 | ralrimiva 2949 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦)) |
9 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥-∞ | |
10 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥 < | |
11 | pimgtmnf2.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
12 | nfcv 2751 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
13 | 11, 12 | nffv 6110 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
14 | 9, 10, 13 | nfbr 4629 | . . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦) |
15 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥) | |
16 | fveq2 6103 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) | |
17 | 16 | breq2d 4595 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (-∞ < (𝐹‘𝑦) ↔ -∞ < (𝐹‘𝑥))) |
18 | 14, 15, 17 | cbvral 3143 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦) ↔ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)) |
19 | 8, 18 | sylib 207 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)) |
20 | 4, 19 | jca 553 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))) |
21 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
22 | 21, 21 | ssrabf 38329 | . . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))) |
23 | 20, 22 | sylibr 223 | . 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)}) |
24 | 2, 23 | eqssd 3585 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Ⅎwnfc 2738 ∀wral 2896 {crab 2900 ⊆ wss 3540 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 ℝcr 9814 -∞cmnf 9951 < clt 9953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 |
This theorem is referenced by: pimgtmnf 39609 |
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