Theorem resdisj 5086

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 4948	. 2 |- ( ( C |` A ) |` B ) = ( C |` ( A i^i B ) )
2		reseq2 4931	. . 3 |- ( ( A i^i B ) = (/) -> ( C |` ( A i^i B ) ) = ( C |` (/) ) )
3		res0 4940	. . 3 |- ( C |` (/) ) = (/)
4	2, 3	eqtrdi 2242	. 2 |- ( ( A i^i B ) = (/) -> ( C |` ( A i^i B ) ) = (/) )
5	1, 4	eqtrid 2238	1 |- ( ( A i^i B ) = (/) -> ( ( C |` A ) |` B ) = (/) )

Syntax hints:   -> wi 4   = wceq 1364   i^i cin 3152   (/)c0 3446   |` cres 4657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091  df-xp 4661  df-rel 4662  df-res 4667

This theorem is referenced by:  fvsnun1  5747