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Mirrors > Home > MPE Home > Th. List > Mathboxes > clsneibex | Structured version Visualization version GIF version |
Description: If (pseudo-)closure and (pseudo-)neighborhood functions are related by the composite operator, 𝐻, then the base set exists. (Contributed by RP, 4-Jun-2021.) |
Ref | Expression |
---|---|
clsneibex.d | ⊢ 𝐷 = (𝑃‘𝐵) |
clsneibex.h | ⊢ 𝐻 = (𝐹 ∘ 𝐷) |
clsneibex.r | ⊢ (𝜑 → 𝐾𝐻𝑁) |
Ref | Expression |
---|---|
clsneibex | ⊢ (𝜑 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clsneibex.h | . . . . 5 ⊢ 𝐻 = (𝐹 ∘ 𝐷) | |
2 | clsneibex.d | . . . . . 6 ⊢ 𝐷 = (𝑃‘𝐵) | |
3 | 2 | coeq2i 5730 | . . . . 5 ⊢ (𝐹 ∘ 𝐷) = (𝐹 ∘ (𝑃‘𝐵)) |
4 | 1, 3 | eqtri 2844 | . . . 4 ⊢ 𝐻 = (𝐹 ∘ (𝑃‘𝐵)) |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐻 = (𝐹 ∘ (𝑃‘𝐵))) |
6 | clsneibex.r | . . 3 ⊢ (𝜑 → 𝐾𝐻𝑁) | |
7 | 5, 6 | breqdi 5080 | . 2 ⊢ (𝜑 → 𝐾(𝐹 ∘ (𝑃‘𝐵))𝑁) |
8 | brne0 5115 | . 2 ⊢ (𝐾(𝐹 ∘ (𝑃‘𝐵))𝑁 → (𝐹 ∘ (𝑃‘𝐵)) ≠ ∅) | |
9 | fvprc 6662 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → (𝑃‘𝐵) = ∅) | |
10 | 9 | rneqd 5807 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ V → ran (𝑃‘𝐵) = ran ∅) |
11 | rn0 5795 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
12 | 10, 11 | syl6eq 2872 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → ran (𝑃‘𝐵) = ∅) |
13 | 12 | ineq2d 4188 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃‘𝐵)) = (dom 𝐹 ∩ ∅)) |
14 | in0 4344 | . . . . 5 ⊢ (dom 𝐹 ∩ ∅) = ∅ | |
15 | 13, 14 | syl6eq 2872 | . . . 4 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃‘𝐵)) = ∅) |
16 | 15 | coemptyd 14338 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐹 ∘ (𝑃‘𝐵)) = ∅) |
17 | 16 | necon1ai 3043 | . 2 ⊢ ((𝐹 ∘ (𝑃‘𝐵)) ≠ ∅ → 𝐵 ∈ V) |
18 | 7, 8, 17 | 3syl 18 | 1 ⊢ (𝜑 → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∩ cin 3934 ∅c0 4290 class class class wbr 5065 dom cdm 5554 ran crn 5555 ∘ ccom 5558 ‘cfv 6354 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-iota 6313 df-fv 6362 |
This theorem is referenced by: clsneircomplex 40451 clsneif1o 40452 clsneicnv 40453 clsneikex 40454 clsneinex 40455 clsneiel1 40456 |
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