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Mirrors > Home > MPE Home > Th. List > Mathboxes > br1cossincnvepres | Structured version Visualization version GIF version |
Description: 𝐵 and 𝐶 are cosets by an intersection with the restricted converse epsilon class: a binary relation. (Contributed by Peter Mazsa, 31-Dec-2021.) |
Ref | Expression |
---|---|
br1cossincnvepres | ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | br1cossinres 35681 | . 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶)))) | |
2 | brcnvep 35520 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐵 ↔ 𝐵 ∈ 𝑢)) | |
3 | 2 | elv 3500 | . . . . 5 ⊢ (𝑢◡ E 𝐵 ↔ 𝐵 ∈ 𝑢) |
4 | 3 | anbi1i 625 | . . . 4 ⊢ ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ↔ (𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵)) |
5 | brcnvep 35520 | . . . . . 6 ⊢ (𝑢 ∈ V → (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢)) | |
6 | 5 | elv 3500 | . . . . 5 ⊢ (𝑢◡ E 𝐶 ↔ 𝐶 ∈ 𝑢) |
7 | 6 | anbi1i 625 | . . . 4 ⊢ ((𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶) ↔ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶)) |
8 | 4, 7 | anbi12i 628 | . . 3 ⊢ (((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶))) |
9 | 8 | rexbii 3247 | . 2 ⊢ (∃𝑢 ∈ 𝐴 ((𝑢◡ E 𝐵 ∧ 𝑢𝑅𝐵) ∧ (𝑢◡ E 𝐶 ∧ 𝑢𝑅𝐶)) ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶))) |
10 | 1, 9 | syl6bb 289 | 1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵 ≀ (𝑅 ∩ (◡ E ↾ 𝐴))𝐶 ↔ ∃𝑢 ∈ 𝐴 ((𝐵 ∈ 𝑢 ∧ 𝑢𝑅𝐵) ∧ (𝐶 ∈ 𝑢 ∧ 𝑢𝑅𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∃wrex 3139 Vcvv 3495 ∩ cin 3935 class class class wbr 5059 E cep 5459 ◡ccnv 5549 ↾ cres 5552 ≀ ccoss 35447 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-eprel 5460 df-xp 5556 df-rel 5557 df-cnv 5558 df-res 5562 df-coss 35653 |
This theorem is referenced by: (None) |
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