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Mirrors > Home > MPE Home > Th. List > Mathboxes > opnneilv | Structured version Visualization version GIF version |
Description: The converse of opnneir 48703 with different dummy variables. Note that the second hypothesis could be generalized by adding 𝑦 ∈ 𝐽 to the antecedent. See the proof for details. Although 𝐽 ∈ Top might be redundant here (see neircl 48701), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024.) |
Ref | Expression |
---|---|
opnneir.1 | ⊢ (𝜑 → 𝐽 ∈ Top) |
opnneilv.2 | ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
opnneilv | ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3069 | . 2 ⊢ (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 ↔ ∃𝑥(𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓)) | |
2 | opnneir.1 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top) | |
3 | neii2 23132 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥)) | |
4 | 2, 3 | sylan 580 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥)) |
5 | 4 | r19.41dv 48651 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓)) |
6 | 5 | expl 457 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓))) |
7 | anass 468 | . . . . . 6 ⊢ (((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) ↔ (𝑆 ⊆ 𝑦 ∧ (𝑦 ⊆ 𝑥 ∧ 𝜓))) | |
8 | opnneilv.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒)) | |
9 | 8 | expimpd 453 | . . . . . . 7 ⊢ (𝜑 → ((𝑦 ⊆ 𝑥 ∧ 𝜓) → 𝜒)) |
10 | 9 | anim2d 612 | . . . . . 6 ⊢ (𝜑 → ((𝑆 ⊆ 𝑦 ∧ (𝑦 ⊆ 𝑥 ∧ 𝜓)) → (𝑆 ⊆ 𝑦 ∧ 𝜒))) |
11 | 7, 10 | biimtrid 242 | . . . . 5 ⊢ (𝜑 → (((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) → (𝑆 ⊆ 𝑦 ∧ 𝜒))) |
12 | 11 | reximdv 3168 | . . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐽 ((𝑆 ⊆ 𝑦 ∧ 𝑦 ⊆ 𝑥) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒))) |
13 | 6, 12 | syld 47 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒))) |
14 | 13 | exlimdv 1931 | . 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ ((nei‘𝐽)‘𝑆) ∧ 𝜓) → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒))) |
15 | 1, 14 | biimtrid 242 | 1 ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 ‘cfv 6563 Topctop 22915 neicnei 23121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-top 22916 df-nei 23122 |
This theorem is referenced by: opnneil 48706 |
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