Theorem resdisj 4975

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 4839	. 2 |- ( ( C |` A ) |` B ) = ( C |` ( A i^i B ) )
2		reseq2 4822	. . 3 |- ( ( A i^i B ) = (/) -> ( C |` ( A i^i B ) ) = ( C |` (/) ) )
3		res0 4831	. . 3 |- ( C |` (/) ) = (/)
4	2, 3	eqtrdi 2189	. 2 |- ( ( A i^i B ) = (/) -> ( C |` ( A i^i B ) ) = (/) )
5	1, 4	syl5eq 2185	1 |- ( ( A i^i B ) = (/) -> ( ( C |` A ) |` B ) = (/) )

Syntax hints:   -> wi 4   = wceq 1332   i^i cin 3075   (/)c0 3368   |` cres 4549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-opab 3998  df-xp 4553  df-rel 4554  df-res 4559

This theorem is referenced by:  fvsnun1  5625