Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > relopabi | GIF version |
Description: A class of ordered pairs is a relation. (Contributed by Mario Carneiro, 21-Dec-2013.) |
Ref | Expression |
---|---|
relopabi.1 | ⊢ 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
Ref | Expression |
---|---|
relopabi | ⊢ Rel 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relopabi.1 | . . . 4 ⊢ 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
2 | df-opab 3990 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
3 | 1, 2 | eqtri 2160 | . . 3 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
4 | vex 2689 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | vex 2689 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | opelvv 4589 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ (V × V) |
7 | eleq1 2202 | . . . . . . 7 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ (V × V) ↔ 〈𝑥, 𝑦〉 ∈ (V × V))) | |
8 | 6, 7 | mpbiri 167 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝑧 ∈ (V × V)) |
9 | 8 | adantr 274 | . . . . 5 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ (V × V)) |
10 | 9 | exlimivv 1868 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ (V × V)) |
11 | 10 | abssi 3172 | . . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ⊆ (V × V) |
12 | 3, 11 | eqsstri 3129 | . 2 ⊢ 𝐴 ⊆ (V × V) |
13 | df-rel 4546 | . 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
14 | 12, 13 | mpbir 145 | 1 ⊢ Rel 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 {cab 2125 Vcvv 2686 ⊆ wss 3071 〈cop 3530 {copab 3988 × cxp 4537 Rel wrel 4544 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-opab 3990 df-xp 4545 df-rel 4546 |
This theorem is referenced by: relopab 4666 mptrel 4667 reli 4668 rele 4669 relcnv 4917 cotr 4920 relco 5037 reloprab 5819 reldmoprab 5856 eqer 6461 ecopover 6527 ecopoverg 6530 relen 6638 reldom 6639 enq0er 7243 aprcl 8408 climrel 11049 brstruct 11968 |
Copyright terms: Public domain | W3C validator |