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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 4067 | . . . 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 2742 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
5 | vex 2742 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | opelvv 4678 | . . . . . . 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 3232 | . . 3 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ⊆ (V × V) |
12 | 3, 11 | eqsstri 3189 | . 2 ⊢ 𝐴 ⊆ (V × V) |
13 | df-rel 4635 | . 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 2739 ⊆ wss 3131 ⟨cop 3597 {copab 4065 × cxp 4626 Rel wrel 4633 |
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 4123 ax-pow 4176 ax-pr 4211 |
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 2741 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-opab 4067 df-xp 4634 df-rel 4635 |
This theorem is referenced by: relopab 4755 mptrel 4757 reli 4758 rele 4759 relcnv 5008 cotr 5012 relco 5129 reloprab 5925 reldmoprab 5962 eqer 6569 ecopover 6635 ecopoverg 6638 relen 6746 reldom 6747 enq0er 7436 aprcl 8605 aptap 8609 climrel 11290 brstruct 12473 |
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