Theorem oprabss 7381

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

Step	Hyp	Ref	Expression
1		reloprab 7334	. . 3 ⊢ Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
2		relssdmrn 6172	. . 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 7380	. . . 4 ⊢ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
5		df-rel 5596	. . . 4 ⊢ (Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
6	4, 5	mpbi 229	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
7		ssv 3945	. . 3 ⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ V
8		xpss12 5604	. . 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 689	. 2 ⊢ (dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} × ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}) ⊆ ((V × V) × V)
10	3, 9	sstri 3930	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ ((V × V) × V)

Syntax hints:  Vcvv 3432   ⊆ wss 3887   × cxp 5587  dom cdm 5589  ran crn 5590  Rel wrel 5594  {coprab 7276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-oprab 7279

This theorem is referenced by:  elmpps  33535