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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pimltpnf2f | Structured version Visualization version GIF version |
Description: Given a real-valued function, the preimage of an open interval, unbounded below, with upper bound +∞, is the whole domain. (Contributed by Glauco Siliprandi, 15-Dec-2024.) |
Ref | Expression |
---|---|
pimltpnf2f.1 | ⊢ Ⅎ𝑥𝐹 |
pimltpnf2f.2 | ⊢ Ⅎ𝑥𝐴 |
pimltpnf2f.3 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
Ref | Expression |
---|---|
pimltpnf2f | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pimltpnf2f.2 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2908 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1913 | . . 3 ⊢ Ⅎ𝑦(𝐹‘𝑥) < +∞ | |
4 | pimltpnf2f.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2908 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
6 | 4, 5 | nffv 6930 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
7 | nfcv 2908 | . . . 4 ⊢ Ⅎ𝑥 < | |
8 | nfcv 2908 | . . . 4 ⊢ Ⅎ𝑥+∞ | |
9 | 6, 7, 8 | nfbr 5213 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑦) < +∞ |
10 | fveq2 6920 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | breq1d 5176 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < +∞ ↔ (𝐹‘𝑦) < +∞)) |
12 | 1, 2, 3, 9, 11 | cbvrabw 3481 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} |
13 | nfv 1913 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
14 | pimltpnf2f.3 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
15 | 14 | ffvelcdmda 7118 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
16 | 13, 15 | pimltpnf 46625 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} = 𝐴) |
17 | 12, 16 | eqtrid 2792 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 Ⅎwnfc 2893 {crab 3443 class class class wbr 5166 ⟶wf 6569 ‘cfv 6573 ℝcr 11183 +∞cpnf 11321 < clt 11324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-fv 6581 df-pnf 11326 df-xr 11328 df-ltxr 11329 |
This theorem is referenced by: pimltpnf2 46634 smfpimltxr 46668 |
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