Theorem relrelss 6294

Description: Two ways to describe the structure of a two-place operation. (Contributed by NM, 17-Dec-2008.)

Assertion

Ref	Expression
relrelss	⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))

Proof of Theorem relrelss

Step	Hyp	Ref	Expression
1		df-rel 5695	. . 3 ⊢ (Rel dom 𝐴 ↔ dom 𝐴 ⊆ (V × V))
2	1	anbi2i 623	. 2 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
3		relssdmrn 6289	. . . 4 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
4		ssv 4019	. . . . 5 ⊢ ran 𝐴 ⊆ V
5		xpss12 5703	. . . . 5 ⊢ ((dom 𝐴 ⊆ (V × V) ∧ ran 𝐴 ⊆ V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
6	4, 5	mpan2 691	. . . 4 ⊢ (dom 𝐴 ⊆ (V × V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
7	3, 6	sylan9ss 4008	. . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V))
8		xpss 5704	. . . . . 6 ⊢ ((V × V) × V) ⊆ (V × V)
9		sstr 4003	. . . . . 6 ⊢ ((𝐴 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (V × V)) → 𝐴 ⊆ (V × V))
10	8, 9	mpan2 691	. . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ (V × V))
11		df-rel 5695	. . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
12	10, 11	sylibr 234	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → Rel 𝐴)
13		dmss 5915	. . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V))
14		vn0 4350	. . . . . 6 ⊢ V ≠ ∅
15		dmxp 5941	. . . . . 6 ⊢ (V ≠ ∅ → dom ((V × V) × V) = (V × V))
16	14, 15	ax-mp 5	. . . . 5 ⊢ dom ((V × V) × V) = (V × V)
17	13, 16	sseqtrdi 4045	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ (V × V))
18	12, 17	jca 511	. . 3 ⊢ (𝐴 ⊆ ((V × V) × V) → (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
19	7, 18	impbii 209	. 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) ↔ 𝐴 ⊆ ((V × V) × V))
20	2, 19	bitri 275	1 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1536   ≠ wne 2937  Vcvv 3477   ⊆ wss 3962  ∅c0 4338   × cxp 5686  dom cdm 5688  ran crn 5689  Rel wrel 5693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699

This theorem is referenced by:  dftpos3  8267  tpostpos2  8270