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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 7207 | . . 3 ⊢ Rel {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
2 | relssdmrn 6115 | . . 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 7253 | . . . 4 ⊢ Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | df-rel 5556 | . . . 4 ⊢ (Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (V × V)) | |
6 | 4, 5 | mpbi 232 | . . 3 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (V × V) |
7 | ssv 3990 | . . 3 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ V | |
8 | xpss12 5564 | . . 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 690 | . 2 ⊢ (dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} × ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) ⊆ ((V × V) × V) |
10 | 3, 9 | sstri 3975 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ ((V × V) × V) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3494 ⊆ wss 3935 × cxp 5547 dom cdm 5549 ran crn 5550 Rel wrel 5554 {coprab 7151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 df-oprab 7154 |
This theorem is referenced by: elmpps 32815 |
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