Theorem relrelss 6234

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 5638	. . 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 6229	. . . 4 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
4		ssv 3968	. . . . 5 ⊢ ran 𝐴 ⊆ V
5		xpss12 5646	. . . . 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 3957	. . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V))
8		xpss 5647	. . . . . 6 ⊢ ((V × V) × V) ⊆ (V × V)
9		sstr 3952	. . . . . 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 5638	. . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
12	10, 11	sylibr 234	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → Rel 𝐴)
13		dmss 5856	. . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V))
14		vn0 4304	. . . . . 6 ⊢ V ≠ ∅
15		dmxp 5882	. . . . . 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 3984	. . . 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 1540   ≠ wne 2925  Vcvv 3444   ⊆ wss 3911  ∅c0 4292   × cxp 5629  dom cdm 5631  ran crn 5632  Rel wrel 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by:  dftpos3  8200  tpostpos2  8203