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Mathbox for Jeff Madsen |
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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 | relopabiv 5820 | . . . 4 ⊢ Rel 𝑅 |
3 | 2 | brrelex1i 5732 | . . 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 5539 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) |
7 | 6 | biimpd 228 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 → 𝜒)) |
8 | 7 | ex 413 | . . . 4 ⊢ (𝐴 ∈ V → (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → 𝜒))) |
9 | 8 | com3l 89 | . . 3 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → (𝐴 ∈ V → 𝜒))) |
10 | 3, 9 | mpdi 45 | . 2 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → 𝜒)) |
11 | brabg2.4 | . . 3 ⊢ (𝜒 → 𝐴 ∈ 𝐶) | |
12 | 4, 5, 1 | brabg 5539 | . . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) |
13 | 12 | exbiri 809 | . . . 4 ⊢ (𝐴 ∈ 𝐶 → (𝐵 ∈ 𝐷 → (𝜒 → 𝐴𝑅𝐵))) |
14 | 13 | com3l 89 | . . 3 ⊢ (𝐵 ∈ 𝐷 → (𝜒 → (𝐴 ∈ 𝐶 → 𝐴𝑅𝐵))) |
15 | 11, 14 | mpdi 45 | . 2 ⊢ (𝐵 ∈ 𝐷 → (𝜒 → 𝐴𝑅𝐵)) |
16 | 10, 15 | impbid 211 | 1 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 class class class wbr 5148 {copab 5210 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 |
This theorem is referenced by: (None) |
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