| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > brabg | Structured version Visualization version 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 509 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜒)) |
| 4 | brabg.5 | . 2 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 5 | 3, 4 | brabga 5539 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 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-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 |
| This theorem is referenced by: brab 5548 ideqg 5862 brcnvg 5890 f1owe 7373 brrpssg 7745 soseq 8184 breng 8994 brdom2g 8996 brdomgOLD 8998 brwdom 9607 brttrcl 9753 ltprord 11070 shftfib 15111 efgrelexlema 19767 isref 23517 sltval 27692 brsslt 27830 lrrecval 27972 istrkgld 28467 islnopp 28747 axcontlem5 28983 cmbr 31603 leopg 32141 cvbr 32301 mdbr 32313 dmdbr 32318 isfne 36340 brabg2 37724 isriscg 37991 brssr 38502 lcvbr 39022 bropabg 43336 |
| Copyright terms: Public domain | W3C validator |