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Mirrors > Home > MPE Home > Th. List > oprabss | Structured version Visualization version 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 7491 | . . 3 ⊢ Rel {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
2 | relssdmrn 6289 | . . 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 7538 | . . . 4 ⊢ Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | df-rel 5695 | . . . 4 ⊢ (Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (V × V)) | |
6 | 4, 5 | mpbi 230 | . . 3 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (V × V) |
7 | ssv 4019 | . . 3 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ V | |
8 | xpss12 5703 | . . 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 692 | . 2 ⊢ (dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} × ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) ⊆ ((V × V) × V) |
10 | 3, 9 | sstri 4004 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ ((V × V) × V) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3477 ⊆ wss 3962 × cxp 5686 dom cdm 5688 ran crn 5689 Rel wrel 5693 {coprab 7431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-rn 5699 df-oprab 7434 |
This theorem is referenced by: elmpps 35557 |
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