Theorem oprabss 6102

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 6064	. . 3 ⊢ Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
2		relssdmrn 5255	. . 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 6101	. . . 4 ⊢ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
5		df-rel 4730	. . . 4 ⊢ (Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ (V × V))
6	4, 5	mpbi 145	. . 3 ⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ (V × V)
7		ssv 3247	. . 3 ⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ V
8		xpss12 4831	. . 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 426	. 2 ⊢ (dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} × ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}) ⊆ ((V × V) × V)
10	3, 9	sstri 3234	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ ((V × V) × V)

Syntax hints:  Vcvv 2800   ⊆ wss 3198   × cxp 4721  dom cdm 4723  ran crn 4724  Rel wrel 4728  {coprab 6014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-dm 4733  df-rn 4734  df-oprab 6017

This theorem is referenced by: (None)