Mathbox for Glauco Siliprandi |
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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 2764 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1843 | . . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑥) < +∞ | |
4 | pimltpnf2.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑥𝑦 | |
6 | 4, 5 | nffv 6198 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑦) |
7 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑥 < | |
8 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑥+∞ | |
9 | 6, 7, 8 | nfbr 4699 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑦) < +∞ |
10 | fveq2 6191 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦)) | |
11 | 10 | breq1d 4663 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑥) < +∞ ↔ (𝐹‘𝑦) < +∞)) |
12 | 1, 2, 3, 9, 11 | cbvrab 3198 | . . 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 1843 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
15 | pimltpnf2.2 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
16 | 15 | ffvelrnda 6359 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ ℝ) |
17 | 14, 16 | pimltpnf 40916 | . 2 ⊢ (𝜑 → {𝑦 ∈ 𝐴 ∣ (𝐹‘𝑦) < +∞} = 𝐴) |
18 | 13, 17 | eqtrd 2656 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) < +∞} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 Ⅎwnfc 2751 {crab 2916 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 ℝcr 9935 +∞cpnf 10071 < clt 10074 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-pnf 10076 df-xr 10078 df-ltxr 10079 |
This theorem is referenced by: smfpimltxr 40956 |
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