Theorem resdisj 3471

Description: A double restriction to disjoint classes is the empty set.

Assertion

Ref	Expression
resdisj	|- ((A i^i B) = (/) -> ((C |` A) |` B) = (/))

Proof of Theorem resdisj

Step	Hyp	Ref	Expression
1		xpdisj1 3468	. . . 4 |- ((A i^i B) = (/) -> ((A X. V) i^i (B X. V)) = (/))
2	1	ineq2d 2217	. . 3 |- ((A i^i B) = (/) -> (C i^i ((A X. V) i^i (B X. V))) = (C i^i (/)))
3		in0 2298	. . 3 |- (C i^i (/)) = (/)
4	2, 3	syl6eq 1523	. 2 |- ((A i^i B) = (/) -> (C i^i ((A X. V) i^i (B X. V))) = (/))
5		df-res 3190	. . 3 |- ((C |` A) |` B) = ((C |` A) i^i (B X. V))
6		df-res 3190	. . . 4 |- (C |` A) = (C i^i (A X. V))
7	6	ineq1i 2213	. . 3 |- ((C |` A) i^i (B X. V)) = ((C i^i (A X. V)) i^i (B X. V))
8		inass 2223	. . 3 |- ((C i^i (A X. V)) i^i (B X. V)) = (C i^i ((A X. V) i^i (B X. V)))
9	5, 7, 8	3eqtr 1499	. 2 |- ((C |` A) |` B) = (C i^i ((A X. V) i^i (B X. V)))
10	4, 9	syl5eq 1519	1 |- ((A i^i B) = (/) -> ((C |` A) |` B) = (/))

Syntax hints:   -> wi 3   = wceq 956  Vcvv 1811   i^i cin 2046  (/)c0 2280   X. cxp 3168   |` cres 3172

This theorem is referenced by:  fvsnun1 3795

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-rel 3185  df-res 3190

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