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Mirrors > Home > ILE Home > Th. List > brab | GIF version |
Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) |
Ref | Expression |
---|---|
opelopab.1 | ⊢ 𝐴 ∈ V |
opelopab.2 | ⊢ 𝐵 ∈ V |
opelopab.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
opelopab.4 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
brab.5 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
Ref | Expression |
---|---|
brab | ⊢ (𝐴𝑅𝐵 ↔ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelopab.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opelopab.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | opelopab.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | opelopab.4 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
5 | brab.5 | . . 3 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
6 | 3, 4, 5 | brabg 4242 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒)) |
7 | 1, 2, 6 | mp2an 423 | 1 ⊢ (𝐴𝑅𝐵 ↔ 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1342 ∈ wcel 2135 Vcvv 2722 class class class wbr 3977 {copab 4037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2724 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-br 3978 df-opab 4039 |
This theorem is referenced by: dftpos4 6223 enq0sym 7365 enq0ref 7366 enq0tr 7367 shftfn 10756 |
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