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| Mirrors > Home > MPE Home > Th. List > braba | Structured version Visualization version GIF version | ||
| Description: The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.) |
| Ref | Expression |
|---|---|
| opelopaba.1 | ⊢ 𝐴 ∈ V |
| opelopaba.2 | ⊢ 𝐵 ∈ V |
| opelopaba.3 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
| braba.4 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
| Ref | Expression |
|---|---|
| braba | ⊢ (𝐴𝑅𝐵 ↔ 𝜓) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opelopaba.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | opelopaba.2 | . 2 ⊢ 𝐵 ∈ V | |
| 3 | opelopaba.3 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
| 4 | braba.4 | . . 3 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 5 | 3, 4 | brabga 5539 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜓)) |
| 6 | 1, 2, 5 | mp2an 692 | 1 ⊢ (𝐴𝑅𝐵 ↔ 𝜓) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 {copab 5205 |
| 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-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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 |
| This theorem is referenced by: frgpuplem 19790 2ndcctbss 23463 legov 28593 prtlem13 38869 wepwsolem 43054 fnwe2val 43061 grumnud 44305 sprsymrelf 47482 catprsc 48902 catprsc2 48903 |
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