Theorem relrelss 5270

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 4738	. . 3 ⊢ (Rel dom 𝐴 ↔ dom 𝐴 ⊆ (V × V))
2	1	anbi2i 457	. 2 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
3		relssdmrn 5264	. . . 4 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
4		ssv 3250	. . . . 5 ⊢ ran 𝐴 ⊆ V
5		xpss12 4839	. . . . 5 ⊢ ((dom 𝐴 ⊆ (V × V) ∧ ran 𝐴 ⊆ V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
6	4, 5	mpan2 425	. . . 4 ⊢ (dom 𝐴 ⊆ (V × V) → (dom 𝐴 × ran 𝐴) ⊆ ((V × V) × V))
7	3, 6	sylan9ss 3241	. . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V))
8		xpss 4840	. . . . . 6 ⊢ ((V × V) × V) ⊆ (V × V)
9		sstr 3236	. . . . . 6 ⊢ ((𝐴 ⊆ ((V × V) × V) ∧ ((V × V) × V) ⊆ (V × V)) → 𝐴 ⊆ (V × V))
10	8, 9	mpan2 425	. . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → 𝐴 ⊆ (V × V))
11		df-rel 4738	. . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
12	10, 11	sylibr 134	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → Rel 𝐴)
13		dmss 4936	. . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V))
14		vn0m 3508	. . . . . 6 ⊢ ∃𝑥 𝑥 ∈ V
15		dmxpm 4958	. . . . . 6 ⊢ (∃𝑥 𝑥 ∈ V → dom ((V × V) × V) = (V × V))
16	14, 15	ax-mp 5	. . . . 5 ⊢ dom ((V × V) × V) = (V × V)
17	13, 16	sseqtrdi 3276	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ (V × V))
18	12, 17	jca 306	. . 3 ⊢ (𝐴 ⊆ ((V × V) × V) → (Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)))
19	7, 18	impbii 126	. 2 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) ↔ 𝐴 ⊆ ((V × V) × V))
20	2, 19	bitri 184	1 ⊢ ((Rel 𝐴 ∧ Rel dom 𝐴) ↔ 𝐴 ⊆ ((V × V) × V))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803   ⊆ wss 3201   × cxp 4729  dom cdm 4731  ran crn 4732  Rel wrel 4736

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by:  dftpos3  6471  tpostpos2  6474