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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 5573 | . . 3 ⊢ Rel {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
2 | relssdmrn 4861 | . . 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 7 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} × ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
4 | reldmoprab 5609 | . . . 4 ⊢ Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | df-rel 4370 | . . . 4 ⊢ (Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (V × V)) | |
6 | 4, 5 | mpbi 143 | . . 3 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (V × V) |
7 | ssv 3019 | . . 3 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ V | |
8 | xpss12 4463 | . . 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 416 | . 2 ⊢ (dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} × ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) ⊆ ((V × V) × V) |
10 | 3, 9 | sstri 3008 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ ((V × V) × V) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2601 ⊆ wss 2973 × cxp 4361 dom cdm 4363 ran crn 4364 Rel wrel 4368 {coprab 5533 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 df-oprab 5536 |
This theorem is referenced by: (None) |
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