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Mirrors > Home > MPE Home > Th. List > Mathboxes > neicvgbex | Structured version Visualization version GIF version |
Description: If (pseudo-)neighborhood and (pseudo-)convergent functions are related by the composite operator, 𝐻, then the base set exists. (Contributed by RP, 4-Jun-2021.) |
Ref | Expression |
---|---|
neicvgbex.d | ⊢ 𝐷 = (𝑃‘𝐵) |
neicvgbex.h | ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) |
neicvgbex.r | ⊢ (𝜑 → 𝑁𝐻𝑀) |
Ref | Expression |
---|---|
neicvgbex | ⊢ (𝜑 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neicvgbex.h | . . . . 5 ⊢ 𝐻 = (𝐹 ∘ (𝐷 ∘ 𝐺)) | |
2 | neicvgbex.d | . . . . . . 7 ⊢ 𝐷 = (𝑃‘𝐵) | |
3 | 2 | coeq1i 5723 | . . . . . 6 ⊢ (𝐷 ∘ 𝐺) = ((𝑃‘𝐵) ∘ 𝐺) |
4 | 3 | coeq2i 5724 | . . . . 5 ⊢ (𝐹 ∘ (𝐷 ∘ 𝐺)) = (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) |
5 | 1, 4 | eqtri 2841 | . . . 4 ⊢ 𝐻 = (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) |
6 | 5 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐻 = (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))) |
7 | neicvgbex.r | . . 3 ⊢ (𝜑 → 𝑁𝐻𝑀) | |
8 | 6, 7 | breqdi 5072 | . 2 ⊢ (𝜑 → 𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀) |
9 | brne0 5107 | . 2 ⊢ (𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀 → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅) | |
10 | fvprc 6656 | . . . . . . . . . . . . 13 ⊢ (¬ 𝐵 ∈ V → (𝑃‘𝐵) = ∅) | |
11 | 10 | dmeqd 5767 | . . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = dom ∅) |
12 | dm0 5783 | . . . . . . . . . . . 12 ⊢ dom ∅ = ∅ | |
13 | 11, 12 | syl6eq 2869 | . . . . . . . . . . 11 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = ∅) |
14 | 13 | ineq1d 4185 | . . . . . . . . . 10 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = (∅ ∩ ran 𝐺)) |
15 | 0in 4344 | . . . . . . . . . 10 ⊢ (∅ ∩ ran 𝐺) = ∅ | |
16 | 14, 15 | syl6eq 2869 | . . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = ∅) |
17 | 16 | coemptyd 14327 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → ((𝑃‘𝐵) ∘ 𝐺) = ∅) |
18 | 17 | rneqd 5801 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ran ∅) |
19 | rn0 5789 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
20 | 18, 19 | syl6eq 2869 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ∅) |
21 | 20 | ineq2d 4186 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = (dom 𝐹 ∩ ∅)) |
22 | in0 4342 | . . . . 5 ⊢ (dom 𝐹 ∩ ∅) = ∅ | |
23 | 21, 22 | syl6eq 2869 | . . . 4 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = ∅) |
24 | 23 | coemptyd 14327 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) = ∅) |
25 | 24 | necon1ai 3040 | . 2 ⊢ ((𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅ → 𝐵 ∈ V) |
26 | 8, 9, 25 | 3syl 18 | 1 ⊢ (𝜑 → 𝐵 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 Vcvv 3492 ∩ cin 3932 ∅c0 4288 class class class wbr 5057 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 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 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: neicvgrcomplex 40341 neicvgf1o 40342 neicvgnvo 40343 neicvgmex 40345 neicvgel1 40347 |
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