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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 3998 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
3 | 1, 2 | eqtri 2161 | . . 3 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
4 | vex 2692 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | vex 2692 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | opelvv 4597 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ (V × V) |
7 | eleq1 2203 | . . . . . . 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 1869 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ (V × V)) |
11 | 10 | abssi 3177 | . . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ⊆ (V × V) |
12 | 3, 11 | eqsstri 3134 | . 2 ⊢ 𝐴 ⊆ (V × V) |
13 | df-rel 4554 | . 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 1332 ∃wex 1469 ∈ wcel 1481 {cab 2126 Vcvv 2689 ⊆ wss 3076 〈cop 3535 {copab 3996 × cxp 4545 Rel wrel 4552 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 df-xp 4553 df-rel 4554 |
This theorem is referenced by: relopab 4674 mptrel 4675 reli 4676 rele 4677 relcnv 4925 cotr 4928 relco 5045 reloprab 5827 reldmoprab 5864 eqer 6469 ecopover 6535 ecopoverg 6538 relen 6646 reldom 6647 enq0er 7267 aprcl 8432 climrel 11081 brstruct 12007 |
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