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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 3810 | . . . 4 ⊢ {〈x, y〉 ∣ φ} = {z ∣ ∃x∃y(z = 〈x, y〉 ∧ φ)} | |
3 | 1, 2 | eqtri 2057 | . . 3 ⊢ A = {z ∣ ∃x∃y(z = 〈x, y〉 ∧ φ)} |
4 | vex 2554 | . . . . . . . 8 ⊢ x ∈ V | |
5 | vex 2554 | . . . . . . . 8 ⊢ y ∈ V | |
6 | 4, 5 | opelvv 4333 | . . . . . . 7 ⊢ 〈x, y〉 ∈ (V × V) |
7 | eleq1 2097 | . . . . . . 7 ⊢ (z = 〈x, y〉 → (z ∈ (V × V) ↔ 〈x, y〉 ∈ (V × V))) | |
8 | 6, 7 | mpbiri 157 | . . . . . 6 ⊢ (z = 〈x, y〉 → z ∈ (V × V)) |
9 | 8 | adantr 261 | . . . . 5 ⊢ ((z = 〈x, y〉 ∧ φ) → z ∈ (V × V)) |
10 | 9 | exlimivv 1773 | . . . 4 ⊢ (∃x∃y(z = 〈x, y〉 ∧ φ) → z ∈ (V × V)) |
11 | 10 | abssi 3009 | . . 3 ⊢ {z ∣ ∃x∃y(z = 〈x, y〉 ∧ φ)} ⊆ (V × V) |
12 | 3, 11 | eqsstri 2969 | . 2 ⊢ A ⊆ (V × V) |
13 | df-rel 4295 | . 2 ⊢ (Rel A ↔ A ⊆ (V × V)) | |
14 | 12, 13 | mpbir 134 | 1 ⊢ Rel A |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1242 ∃wex 1378 ∈ wcel 1390 {cab 2023 Vcvv 2551 ⊆ wss 2911 〈cop 3370 {copab 3808 × cxp 4286 Rel wrel 4293 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-opab 3810 df-xp 4294 df-rel 4295 |
This theorem is referenced by: relopab 4407 reli 4408 rele 4409 relcnv 4646 cotr 4649 relco 4762 reloprab 5495 reldmoprab 5531 eqer 6074 ecopover 6140 ecopoverg 6143 relen 6161 reldom 6162 enq0er 6418 |
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