Theorem relrelss 5155

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 4633	. . 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 5149	. . . 4 ⊢ (Rel 𝐴 → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
4		ssv 3177	. . . . 5 ⊢ ran 𝐴 ⊆ V
5		xpss12 4733	. . . . 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 3168	. . 3 ⊢ ((Rel 𝐴 ∧ dom 𝐴 ⊆ (V × V)) → 𝐴 ⊆ ((V × V) × V))
8		xpss 4734	. . . . . 6 ⊢ ((V × V) × V) ⊆ (V × V)
9		sstr 3163	. . . . . 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 4633	. . . . 5 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
12	10, 11	sylibr 134	. . . 4 ⊢ (𝐴 ⊆ ((V × V) × V) → Rel 𝐴)
13		dmss 4826	. . . . 5 ⊢ (𝐴 ⊆ ((V × V) × V) → dom 𝐴 ⊆ dom ((V × V) × V))
14		vn0m 3434	. . . . . 6 ⊢ ∃𝑥 𝑥 ∈ V
15		dmxpm 4847	. . . . . 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 3203	. . . 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 1353  ∃wex 1492   ∈ wcel 2148  Vcvv 2737   ⊆ wss 3129   × cxp 4624  dom cdm 4626  ran crn 4627  Rel wrel 4631

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-dm 4636  df-rn 4637

This theorem is referenced by:  dftpos3  6262  tpostpos2  6265