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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 3946 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ⊆ 𝐴) |
3 | ssid 3879 | . . . . 5 ⊢ 𝐴 ⊆ 𝐴 | |
4 | 3 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴) |
5 | pimgtmnf2.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
6 | 5 | ffvelrnda 6676 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
7 | 6 | mnfltd 12336 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → -∞ < (𝐹‘𝑦)) |
8 | 7 | ralrimiva 3132 | . . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦)) |
9 | nfcv 2932 | . . . . . . 7 ⊢ Ⅎ𝑥-∞ | |
10 | nfcv 2932 | . . . . . . 7 ⊢ Ⅎ𝑥 < | |
11 | pimgtmnf2.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
12 | nfcv 2932 | . . . . . . . 8 ⊢ Ⅎ𝑥𝑦 | |
13 | 11, 12 | nffv 6509 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
14 | 9, 10, 13 | nfbr 4976 | . . . . . 6 ⊢ Ⅎ𝑥-∞ < (𝐹‘𝑦) |
15 | nfv 1873 | . . . . . 6 ⊢ Ⅎ𝑦-∞ < (𝐹‘𝑥) | |
16 | fveq2 6499 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) | |
17 | 16 | breq2d 4941 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (-∞ < (𝐹‘𝑦) ↔ -∞ < (𝐹‘𝑥))) |
18 | 14, 15, 17 | cbvral 3379 | . . . . 5 ⊢ (∀𝑦 ∈ 𝐴 -∞ < (𝐹‘𝑦) ↔ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)) |
19 | 8, 18 | sylib 210 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥)) |
20 | 4, 19 | jca 504 | . . 3 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))) |
21 | nfcv 2932 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
22 | 21, 21 | ssrabf 40810 | . . 3 ⊢ (𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 -∞ < (𝐹‘𝑥))) |
23 | 20, 22 | sylibr 226 | . 2 ⊢ (𝜑 → 𝐴 ⊆ {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)}) |
24 | 2, 23 | eqssd 3875 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ -∞ < (𝐹‘𝑥)} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 Ⅎwnfc 2916 ∀wral 3088 {crab 3092 ⊆ wss 3829 class class class wbr 4929 ⟶wf 6184 ‘cfv 6188 ℝcr 10334 -∞cmnf 10472 < clt 10474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-fv 6196 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 |
This theorem is referenced by: pimgtmnf 42437 |
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