Theorem relrn0 4891

Description: A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.)

Assertion

Ref	Expression
relrn0	⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))

Proof of Theorem relrn0

Step	Hyp	Ref	Expression
1		reldm0 4847	. 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
2		dm0rn0 4846	. 2 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
3	1, 2	bitrdi 196	1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353  ∅c0 3424  dom cdm 4628  ran crn 4629  Rel wrel 4633

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-in1 614  ax-in2 615  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 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  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-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639

This theorem is referenced by:  cnvsn0  5099