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Mirrors > Home > ILE Home > Th. List > brabg | GIF version |
Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
opelopabg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
opelopabg.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
brabg.5 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
Ref | Expression |
---|---|
brabg | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelopabg.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | opelopabg.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | sylan9bb 462 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜒)) |
4 | brabg.5 | . 2 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
5 | 3, 4 | brabga 4282 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 class class class wbr 4018 {copab 4078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 |
This theorem is referenced by: brab 4290 opbrop 4723 ideqg 4796 opelcnvg 4825 bren 6773 brdomg 6774 enq0breq 7465 ltresr 7868 ltxrlt 8053 apreap 8574 apreim 8590 shftfibg 10861 shftfib 10864 2shfti 10872 |
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