| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > reloprab | Structured version Visualization version GIF version | ||
| Description: An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.) |
| Ref | Expression |
|---|---|
| reloprab | ⊢ Rel {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfoprab2 7491 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑤, 𝑧〉 ∣ ∃𝑥∃𝑦(𝑤 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
| 2 | 1 | relopabiv 5830 | 1 ⊢ Rel {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 〈cop 4632 Rel wrel 5690 {coprab 7432 |
| 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-11 2157 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-opab 5206 df-xp 5691 df-rel 5692 df-oprab 7435 |
| This theorem is referenced by: oprabv 7493 oprabss 7541 brcnvrabga 38343 |
| Copyright terms: Public domain | W3C validator |