![]() |
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 4062 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} | |
3 | 1, 2 | eqtri 2198 | . . 3 ⊢ 𝐴 = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} |
4 | vex 2740 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | vex 2740 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | opelvv 4672 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ∈ (V × V) |
7 | eleq1 2240 | . . . . . . 7 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → (𝑧 ∈ (V × V) ↔ 〈𝑥, 𝑦〉 ∈ (V × V))) | |
8 | 6, 7 | mpbiri 168 | . . . . . 6 ⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝑧 ∈ (V × V)) |
9 | 8 | adantr 276 | . . . . 5 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ (V × V)) |
10 | 9 | exlimivv 1896 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → 𝑧 ∈ (V × V)) |
11 | 10 | abssi 3230 | . . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)} ⊆ (V × V) |
12 | 3, 11 | eqsstri 3187 | . 2 ⊢ 𝐴 ⊆ (V × V) |
13 | df-rel 4629 | . 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
14 | 12, 13 | mpbir 146 | 1 ⊢ Rel 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 {cab 2163 Vcvv 2737 ⊆ wss 3129 〈cop 3594 {copab 4060 × cxp 4620 Rel wrel 4627 |
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 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-opab 4062 df-xp 4628 df-rel 4629 |
This theorem is referenced by: relopab 4749 mptrel 4750 reli 4751 rele 4752 relcnv 5001 cotr 5005 relco 5122 reloprab 5916 reldmoprab 5953 eqer 6560 ecopover 6626 ecopoverg 6629 relen 6737 reldom 6738 enq0er 7412 aprcl 8580 climrel 11259 brstruct 12441 |
Copyright terms: Public domain | W3C validator |