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Mirrors > Home > ILE Home > Th. List > oprabss | GIF version |
Description: Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.) |
Ref | Expression |
---|---|
oprabss | ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ ((V × V) × V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reloprab 5812 | . . 3 ⊢ Rel {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
2 | relssdmrn 5054 | . . 3 ⊢ (Rel {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} × ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑})) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} × ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
4 | reldmoprab 5849 | . . . 4 ⊢ Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | df-rel 4541 | . . . 4 ⊢ (Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (V × V)) | |
6 | 4, 5 | mpbi 144 | . . 3 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (V × V) |
7 | ssv 3114 | . . 3 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ V | |
8 | xpss12 4641 | . . 3 ⊢ ((dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (V × V) ∧ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ V) → (dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} × ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) ⊆ ((V × V) × V)) | |
9 | 6, 7, 8 | mp2an 422 | . 2 ⊢ (dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} × ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) ⊆ ((V × V) × V) |
10 | 3, 9 | sstri 3101 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ ((V × V) × V) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2681 ⊆ wss 3066 × cxp 4532 dom cdm 4534 ran crn 4535 Rel wrel 4539 {coprab 5768 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 df-oprab 5771 |
This theorem is referenced by: (None) |
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