Theorem relrn0 3413

Description: A relation is empty iff its range is empty.

Assertion

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

Proof of Theorem relrn0

Step	Hyp	Ref	Expression
1		reldm0 3388	. 2 ⊢ (Rel A → (A = ∅ ↔ dom A = ∅))
2		dm0rn0 3387	. 2 ⊢ (dom A = ∅ ↔ ran A = ∅)
3	1, 2	syl6bb 547	1 ⊢ (Rel A → (A = ∅ ↔ ran A = ∅))

Syntax hints:   → wi 3   ↔ wb 153   = wceq 997  ∅c0 2331  dom cdm 3227  ran crn 3228  Rel wrel 3232

This theorem is referenced by:  foconst 3740  fconst5 3905  infxpidmlem11 7654

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-pow 2798  ax-pr 2835

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-v 1859  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-br 2675  df-opab 2722  df-xp 3241  df-rel 3242  df-cnv 3243  df-dm 3245  df-rn 3246

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