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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 5870 | . . . . . 6 ⊢ (𝐷 ∘ 𝐺) = ((𝑃‘𝐵) ∘ 𝐺) |
| 4 | 3 | coeq2i 5871 | . . . . 5 ⊢ (𝐹 ∘ (𝐷 ∘ 𝐺)) = (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) |
| 5 | 1, 4 | eqtri 2765 | . . . 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 5158 | . 2 ⊢ (𝜑 → 𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀) |
| 9 | brne0 5193 | . 2 ⊢ (𝑁(𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺))𝑀 → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) ≠ ∅) | |
| 10 | fvprc 6898 | . . . . . . . . . . . . 13 ⊢ (¬ 𝐵 ∈ V → (𝑃‘𝐵) = ∅) | |
| 11 | 10 | dmeqd 5916 | . . . . . . . . . . . 12 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = dom ∅) |
| 12 | dm0 5931 | . . . . . . . . . . . 12 ⊢ dom ∅ = ∅ | |
| 13 | 11, 12 | eqtrdi 2793 | . . . . . . . . . . 11 ⊢ (¬ 𝐵 ∈ V → dom (𝑃‘𝐵) = ∅) |
| 14 | 13 | ineq1d 4219 | . . . . . . . . . 10 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = (∅ ∩ ran 𝐺)) |
| 15 | 0in 4397 | . . . . . . . . . 10 ⊢ (∅ ∩ ran 𝐺) = ∅ | |
| 16 | 14, 15 | eqtrdi 2793 | . . . . . . . . 9 ⊢ (¬ 𝐵 ∈ V → (dom (𝑃‘𝐵) ∩ ran 𝐺) = ∅) |
| 17 | 16 | coemptyd 15018 | . . . . . . . 8 ⊢ (¬ 𝐵 ∈ V → ((𝑃‘𝐵) ∘ 𝐺) = ∅) |
| 18 | 17 | rneqd 5949 | . . . . . . 7 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ran ∅) |
| 19 | rn0 5936 | . . . . . . 7 ⊢ ran ∅ = ∅ | |
| 20 | 18, 19 | eqtrdi 2793 | . . . . . 6 ⊢ (¬ 𝐵 ∈ V → ran ((𝑃‘𝐵) ∘ 𝐺) = ∅) |
| 21 | 20 | ineq2d 4220 | . . . . 5 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = (dom 𝐹 ∩ ∅)) |
| 22 | in0 4395 | . . . . 5 ⊢ (dom 𝐹 ∩ ∅) = ∅ | |
| 23 | 21, 22 | eqtrdi 2793 | . . . 4 ⊢ (¬ 𝐵 ∈ V → (dom 𝐹 ∩ ran ((𝑃‘𝐵) ∘ 𝐺)) = ∅) |
| 24 | 23 | coemptyd 15018 | . . 3 ⊢ (¬ 𝐵 ∈ V → (𝐹 ∘ ((𝑃‘𝐵) ∘ 𝐺)) = ∅) |
| 25 | 24 | necon1ai 2968 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∩ cin 3950 ∅c0 4333 class class class wbr 5143 dom cdm 5685 ran crn 5686 ∘ ccom 5689 ‘cfv 6561 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fv 6569 |
| This theorem is referenced by: neicvgrcomplex 44126 neicvgf1o 44127 neicvgnvo 44128 neicvgmex 44130 neicvgel1 44132 |
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