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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 5802 | . . . . 5 ⊢ (𝐹 ∘ 𝐷) = (𝐹 ∘ (𝑃‘𝐵)) |
4 | 1, 3 | eqtri 2764 | . . . 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 5107 | . 2 ⊢ (𝜑 → 𝐾(𝐹 ∘ (𝑃‘𝐵))𝑁) |
8 | brne0 5142 | . 2 ⊢ (𝐾(𝐹 ∘ (𝑃‘𝐵))𝑁 → (𝐹 ∘ (𝑃‘𝐵)) ≠ ∅) | |
9 | fvprc 6817 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → (𝑃‘𝐵) = ∅) | |
10 | 9 | rneqd 5879 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ V → ran (𝑃‘𝐵) = ran ∅) |
11 | rn0 5867 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
12 | 10, 11 | eqtrdi 2792 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → ran (𝑃‘𝐵) = ∅) |
13 | 12 | ineq2d 4159 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃‘𝐵)) = (dom 𝐹 ∩ ∅)) |
14 | in0 4338 | . . . . 5 ⊢ (dom 𝐹 ∩ ∅) = ∅ | |
15 | 13, 14 | eqtrdi 2792 | . . . 4 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃‘𝐵)) = ∅) |
16 | 15 | coemptyd 14789 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐹 ∘ (𝑃‘𝐵)) = ∅) |
17 | 16 | necon1ai 2968 | . 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 1540 ∈ wcel 2105 ≠ wne 2940 Vcvv 3441 ∩ cin 3897 ∅c0 4269 class class class wbr 5092 dom cdm 5620 ran crn 5621 ∘ ccom 5624 ‘cfv 6479 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-iota 6431 df-fv 6487 |
This theorem is referenced by: clsneircomplex 42034 clsneif1o 42035 clsneicnv 42036 clsneikex 42037 clsneinex 42038 clsneiel1 42039 |
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