Theorem res0 3371

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 3190	. 2 |- (A |` (/)) = (A i^i ((/) X. V))
2		xp0r 3239	. . 3 |- ((/) X. V) = (/)
3	2	ineq2i 2214	. 2 |- (A i^i ((/) X. V)) = (A i^i (/))
4		in0 2298	. 2 |- (A i^i (/)) = (/)
5	1, 3, 4	3eqtr 1499	1 |- (A |` (/)) = (/)

Syntax hints:   = wceq 956  Vcvv 1811   i^i cin 2046  (/)c0 2280   X. cxp 3168   |` cres 3172

This theorem is referenced by:  ima0 3420  tz7.44-1 3928

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-xp 3184  df-res 3190

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