Theorem oprabss 7011

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 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)

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