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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 5724 | . . . . 5 ⊢ (𝐹 ∘ 𝐷) = (𝐹 ∘ (𝑃‘𝐵)) |
4 | 1, 3 | eqtri 2843 | . . . 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 5074 | . 2 ⊢ (𝜑 → 𝐾(𝐹 ∘ (𝑃‘𝐵))𝑁) |
8 | brne0 5109 | . 2 ⊢ (𝐾(𝐹 ∘ (𝑃‘𝐵))𝑁 → (𝐹 ∘ (𝑃‘𝐵)) ≠ ∅) | |
9 | fvprc 6656 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → (𝑃‘𝐵) = ∅) | |
10 | 9 | rneqd 5801 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ V → ran (𝑃‘𝐵) = ran ∅) |
11 | rn0 5789 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
12 | 10, 11 | syl6eq 2871 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → ran (𝑃‘𝐵) = ∅) |
13 | 12 | ineq2d 4182 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃‘𝐵)) = (dom 𝐹 ∩ ∅)) |
14 | in0 4338 | . . . . 5 ⊢ (dom 𝐹 ∩ ∅) = ∅ | |
15 | 13, 14 | syl6eq 2871 | . . . 4 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran (𝑃‘𝐵)) = ∅) |
16 | 15 | coemptyd 14334 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐹 ∘ (𝑃‘𝐵)) = ∅) |
17 | 16 | necon1ai 3042 | . 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 1536 ∈ wcel 2113 ≠ wne 3015 Vcvv 3491 ∩ cin 3928 ∅c0 4284 class class class wbr 5059 dom cdm 5548 ran crn 5549 ∘ ccom 5552 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fv 6356 |
This theorem is referenced by: clsneircomplex 40527 clsneif1o 40528 clsneicnv 40529 clsneikex 40530 clsneinex 40531 clsneiel1 40532 |
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