Theorem resdisj 4872

Description: A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

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

Proof of Theorem resdisj

Step	Hyp	Ref	Expression
1		resres 4738	. 2 |- ( ( C |` A ) |` B ) = ( C |` ( A i^i B ) )
2		reseq2 4721	. . 3 |- ( ( A i^i B ) = (/) -> ( C |` ( A i^i B ) ) = ( C |` (/) ) )
3		res0 4730	. . 3 |- ( C |` (/) ) = (/)
4	2, 3	syl6eq 2137	. 2 |- ( ( A i^i B ) = (/) -> ( C |` ( A i^i B ) ) = (/) )
5	1, 4	syl5eq 2133	1 |- ( ( A i^i B ) = (/) -> ( ( C |` A ) |` B ) = (/) )

Syntax hints:   -> wi 4   = wceq 1290   i^i cin 2999   (/)c0 3287   |` cres 4453

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-opab 3906  df-xp 4457  df-rel 4458  df-res 4463

This theorem is referenced by:  fvsnun1  5508