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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 5830 | . . . 4 ⊢ Rel 𝑅 | 
| 3 | 2 | brrelex1i 5741 | . . 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 5544 | . . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) | 
| 7 | 6 | biimpd 229 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 → 𝜒)) | 
| 8 | 7 | ex 412 | . . . 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 5544 | . . . . 5 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) | 
| 13 | 12 | exbiri 811 | . . . 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 212 | 1 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 ↔ 𝜒)) | 
| 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-ral 3062 df-rex 3071 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 df-xp 5691 df-rel 5692 | 
| This theorem is referenced by: (None) | 
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