Theorem relrelss 6272

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 5683	. . 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 6267	. . . 4 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
4		ssv 4006	. . . . 5 ⊢ ran 𝐴 ⊆ V
5		xpss12 5691	. . . . 5 ⊢ ((dom 𝐴 ⊆ (V × V) ∧ ran 𝐴 ⊆ V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
6	4, 5	mpan2 689	. . . 4 ⊢ (dom 𝐴 ⊆ (V × V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
7	3, 6	sylan9ss 3995	. . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V))
8		xpss 5692	. . . . . 6 ⊢ ((V × V) × V) ⊆ (V × V)
9		sstr 3990	. . . . . 6 ⊢ ((𝐴 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (V × V)) → 𝐴 ⊆ (V × V))
10	8, 9	mpan2 689	. . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ (V × V))
11		df-rel 5683	. . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
12	10, 11	sylibr 233	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → Rel 𝐴)
13		dmss 5902	. . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V))
14		vn0 4338	. . . . . 6 ⊢ V ≠ ∅
15		dmxp 5928	. . . . . 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 4032	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ (V × V))
18	12, 17	jca 512	. . 3 ⊢ (𝐴 ⊆ ((V × V) × V) → (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
19	7, 18	impbii 208	. 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) ↔ 𝐴 ⊆ ((V × V) × V))
20	2, 19	bitri 274	1 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ≠ wne 2940  Vcvv 3474   ⊆ wss 3948  ∅c0 4322   × cxp 5674  dom cdm 5676  ran crn 5677  Rel wrel 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687

This theorem is referenced by:  dftpos3  8228  tpostpos2  8231