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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 4186 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒)) |
7 | 1, 2, 6 | mp2an 422 | 1 ⊢ (𝐴𝑅𝐵 ↔ 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2681 class class class wbr 3924 {copab 3983 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 |
This theorem is referenced by: dftpos4 6153 enq0sym 7233 enq0ref 7234 enq0tr 7235 shftfn 10589 |
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