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Mirrors > Home > MPE Home > Th. List > Mathboxes > brabg2 | Structured version Visualization version GIF version |
Description: Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
brabg2.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
brabg2.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
brabg2.3 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
brabg2.4 | ⊢ (𝜒 → 𝐴 ∈ 𝐶) |
Ref | Expression |
---|---|
brabg2 | ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brabg2.3 | . . . . 5 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | 1 | relopabi 5167 | . . . 4 ⊢ Rel 𝑅 |
3 | 2 | brrelexi 5082 | . . 3 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V) |
4 | brabg2.1 | . . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
5 | brabg2.2 | . . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
6 | 4, 5, 1 | brabg 4919 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) |
7 | 6 | biimpd 218 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 → 𝜒)) |
8 | 7 | ex 449 | . . . 4 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → 𝜒))) |
9 | 8 | com3l 87 | . . 3 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → (𝐴 ∈ V → 𝜒))) |
10 | 3, 9 | mpdi 44 | . 2 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → 𝜒)) |
11 | brabg2.4 | . . 3 ⊢ (𝜒 → 𝐴 ∈ 𝐶) | |
12 | 4, 5, 1 | brabg 4919 | . . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) |
13 | 12 | exbiri 650 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (𝜒 → 𝐴𝑅𝐵))) |
14 | 13 | com3l 87 | . . 3 ⊢ (𝐵 ∈ 𝐷 → (𝜒 → (𝐴 ∈ 𝐶 → 𝐴𝑅𝐵))) |
15 | 11, 14 | mpdi 44 | . 2 ⊢ (𝐵 ∈ 𝐷 → (𝜒 → 𝐴𝑅𝐵)) |
16 | 10, 15 | impbid 201 | 1 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 |
This theorem is referenced by: (None) |
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