Theorem resdisj 3477

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 3474	. . . 4 |- ((A i^i B) = (/) -> ((A X. V) i^i (B X. V)) = (/))
2	1	ineq2d 2220	. . 3 |- ((A i^i B) = (/) -> (C i^i ((A X. V) i^i (B X. V))) = (C i^i (/)))
3		in0 2302	. . 3 |- (C i^i (/)) = (/)
4	2, 3	syl6eq 1526	. 2 |- ((A i^i B) = (/) -> (C i^i ((A X. V) i^i (B X. V))) = (/))
5		df-res 3196	. . 3 |- ((C |` A) |` B) = ((C |` A) i^i (B X. V))
6		df-res 3196	. . . 4 |- (C |` A) = (C i^i (A X. V))
7	6	ineq1i 2216	. . 3 |- ((C |` A) i^i (B X. V)) = ((C i^i (A X. V)) i^i (B X. V))
8		inass 2226	. . 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 1502	. 2 |- ((C |` A) |` B) = (C i^i ((A X. V) i^i (B X. V)))
10	4, 9	syl5eq 1522	1 |- ((A i^i B) = (/) -> ((C |` A) |` B) = (/))

Syntax hints:   -> wi 3   = wceq 958  Vcvv 1814   i^i cin 2049  (/)c0 2283   X. cxp 3174   |` cres 3178

This theorem is referenced by:  fvsnun1 3801

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-xp 3190  df-rel 3191  df-res 3196

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