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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimgtpnf2 | 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 empty set. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Glauco Siliprandi, 15-Dec-2024.) |
Ref | Expression |
---|---|
pimgtpnf2.1 | ⊢ Ⅎ𝑥𝐹 |
pimgtpnf2.2 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
Ref | Expression |
---|---|
pimgtpnf2 | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pimgtpnf2.1 | . 2 ⊢ Ⅎ𝑥𝐹 | |
2 | nfcv 2902 | . 2 ⊢ Ⅎ𝑥𝐴 | |
3 | pimgtpnf2.2 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
4 | 1, 2, 3 | pimgtpnf2f 46660 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ +∞ < (𝐹‘𝑥)} = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 Ⅎwnfc 2887 {crab 3432 ∅c0 4338 class class class wbr 5147 ⟶wf 6558 ‘cfv 6562 ℝcr 11151 +∞cpnf 11289 < clt 11292 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fv 6570 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 |
This theorem is referenced by: (None) |
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