Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > pimltpnf2 | 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, 26-Jun-2021.) |
Ref | Expression |
---|---|
pimltpnf2.1 | ⊢ Ⅎ𝑥𝐹 |
pimltpnf2.2 | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
Ref | Expression |
---|---|
pimltpnf2 | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1830 | . . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑥) < +∞ | |
4 | pimltpnf2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
6 | 4, 5 | nffv 6110 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
7 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥 < | |
8 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥+∞ | |
9 | 6, 7, 8 | nfbr 4629 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) < +∞ |
10 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | breq1d 4593 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < +∞ ↔ (𝐹‘𝑦) < +∞)) |
12 | 1, 2, 3, 9, 11 | cbvrab 3171 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} |
13 | 12 | a1i 11 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞}) |
14 | nfv 1830 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
15 | pimltpnf2.2 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
16 | 15 | ffvelrnda 6267 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
17 | 14, 16 | pimltpnf 39593 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} = 𝐴) |
18 | 13, 17 | eqtrd 2644 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 Ⅎwnfc 2738 {crab 2900 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 ℝcr 9814 +∞cpnf 9950 < 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-xr 9957 df-ltxr 9958 |
This theorem is referenced by: smfpimltxr 39634 |
Copyright terms: Public domain | W3C validator |