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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 6967 | . . 3 ⊢ Rel {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
2 | relssdmrn 5901 | . . 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 7010 | . . . 4 ⊢ Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
5 | df-rel 5353 | . . . 4 ⊢ (Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (V × V)) | |
6 | 4, 5 | mpbi 222 | . . 3 ⊢ dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ (V × V) |
7 | ssv 3850 | . . 3 ⊢ ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ V | |
8 | xpss12 5361 | . . 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 683 | . 2 ⊢ (dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} × ran {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) ⊆ ((V × V) × V) |
10 | 3, 9 | sstri 3836 | 1 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ⊆ ((V × V) × V) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3414 ⊆ wss 3798 × cxp 5344 dom cdm 5346 ran crn 5347 Rel wrel 5351 {coprab 6911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-xp 5352 df-rel 5353 df-cnv 5354 df-dm 5356 df-rn 5357 df-oprab 6914 |
This theorem is referenced by: elmpps 32012 |
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