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Theorem oprabss 6788
Description: Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.)
Assertion
Ref Expression
oprabss {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ ((V × V) × V)
Distinct variable group:   𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem oprabss
StepHypRef Expression
1 reloprab 6744 . . 3 Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
2 relssdmrn 5694 . . 3 (Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ (dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} × ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}))
31, 2ax-mp 5 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ (dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} × ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
4 reldmoprab 6787 . . . 4 Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
5 df-rel 5150 . . . 4 (Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
64, 5mpbi 220 . . 3 dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
7 ssv 3658 . . 3 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ V
8 xpss12 5158 . . 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))
96, 7, 8mp2an 708 . 2 (dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} × ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}) ⊆ ((V × V) × V)
103, 9sstri 3645 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ ((V × V) × V)
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3231  wss 3607   × cxp 5141  dom cdm 5143  ran crn 5144  Rel wrel 5148  {coprab 6691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-oprab 6694
This theorem is referenced by:  elmpps  31596
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