Theorem rn0 2567

Description: The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36.

Assertion

Ref	Expression
rn0	⊢ ran ∅ = ∅

Proof of Theorem rn0

Step	Hyp	Ref	Expression
1		dm0 2542	. 2 ⊢ dom ∅ = ∅
2		dm0rn0 2549	. 2 ⊢ (dom ∅ = ∅ ↔ ran ∅ = ∅)
3	1, 2	mpbi 164	1 ⊢ ran ∅ = ∅

Syntax hints:   = wceq 1091  ∅c0 1707  dom cdm 2410  ran crn 2411

This theorem is referenced by:  ima0 2615  f0 2772  2ndval 3090  map0e 3266  aceq5lem3 3560  fodomb 3615  infxpidmlem4 4936  infxpidmlem8 4940  infxpidmlem10 4942

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429

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