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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 5282 | . . . . 5 ⊢ (𝐹 ∘ 𝐷) = (𝐹 ∘ (𝑃‘𝐵)) |
4 | 1, 3 | eqtri 2644 | . . . 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 4668 | . 2 ⊢ (𝜑 → 𝐾(𝐹 ∘ (𝑃‘𝐵))𝑁) |
8 | brne0 4702 | . 2 ⊢ (𝐾(𝐹 ∘ (𝑃‘𝐵))𝑁 → (𝐹 ∘ (𝑃‘𝐵)) ≠ ∅) | |
9 | fvprc 6185 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → (𝑃‘𝐵) = ∅) | |
10 | 9 | rneqd 5353 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ V → ran (𝑃‘𝐵) = ran ∅) |
11 | rn0 5377 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
12 | 10, 11 | syl6eq 2672 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → ran (𝑃‘𝐵) = ∅) |
13 | 12 | ineq2d 3814 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃‘𝐵)) = (dom 𝐹 ∩ ∅)) |
14 | in0 3968 | . . . . 5 ⊢ (dom 𝐹 ∩ ∅) = ∅ | |
15 | 13, 14 | syl6eq 2672 | . . . 4 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃‘𝐵)) = ∅) |
16 | 15 | coemptyd 13718 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐹 ∘ (𝑃‘𝐵)) = ∅) |
17 | 16 | necon1ai 2821 | . 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 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∩ cin 3573 ∅c0 3915 class class class wbr 4653 dom cdm 5114 ran crn 5115 ∘ ccom 5118 ‘cfv 5888 |
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 |
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-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fv 5896 |
This theorem is referenced by: clsneircomplex 38401 clsneif1o 38402 clsneicnv 38403 clsneikex 38404 clsneinex 38405 clsneiel1 38406 |
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