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Mirrors > Home > MPE Home > Th. List > brcodir | Structured version Visualization version GIF version |
Description: Two ways of saying that two elements have an upper bound. (Contributed by Mario Carneiro, 3-Nov-2015.) |
Ref | Expression |
---|---|
brcodir | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brcog 5701 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵))) | |
2 | vex 3444 | . . . . . 6 ⊢ 𝑧 ∈ V | |
3 | brcnvg 5714 | . . . . . 6 ⊢ ((𝑧 ∈ V ∧ 𝐵 ∈ 𝑊) → (𝑧◡𝑅𝐵 ↔ 𝐵𝑅𝑧)) | |
4 | 2, 3 | mpan 689 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → (𝑧◡𝑅𝐵 ↔ 𝐵𝑅𝑧)) |
5 | 4 | anbi2d 631 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → ((𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ (𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧))) |
6 | 5 | adantl 485 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ (𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧))) |
7 | 6 | exbidv 1922 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑧(𝐴𝑅𝑧 ∧ 𝑧◡𝑅𝐵) ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧))) |
8 | 1, 7 | bitrd 282 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(◡𝑅 ∘ 𝑅)𝐵 ↔ ∃𝑧(𝐴𝑅𝑧 ∧ 𝐵𝑅𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 ◡ccnv 5518 ∘ ccom 5523 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-cnv 5527 df-co 5528 |
This theorem is referenced by: codir 5947 |
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