Theorem res0 2578

Description: A restriction to the empty set is empty.

Assertion

Ref	Expression
res0	|- (A |` (/)) = (/)

Proof of Theorem res0

Step	Hyp	Ref	Expression
1		df-res 2430	. 2 |- (A |` (/)) = (A i^i ((/) X. V))
2		xp0r 2474	. . 3 |- ((/) X. V) = (/)
3	2	ineq2i 1642	. 2 |- (A i^i ((/) X. V)) = (A i^i (/))
4		in0 1722	. 2 |- (A i^i (/)) = (/)
5	1, 3, 4	3eqtr 1123	1 |- (A |` (/)) = (/)

Syntax hints:   = wceq 1091  Vcvv 1348   i^i cin 1486  (/)c0 1707   X. cxp 2408   |` cres 2412

This theorem is referenced by:  ima0 2615  tz7.44-1 2966

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-opab 2098  df-xp 2424  df-res 2430

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