Theorem ssres2 3392

Description: Subclass theorem for restriction.

Assertion

Ref	Expression
ssres2	|- (A (_ B -> (C |` A) (_ (C |` B))

Proof of Theorem ssres2

Step	Hyp	Ref	Expression
1		ssid 2083	. . . 4 |- V (_ V
2		ssxp 3262	. . . 4 |- ((A (_ B /\ V (_ V) -> (A X. V) (_ (B X. V))
3	1, 2	mpan2 698	. . 3 |- (A (_ B -> (A X. V) (_ (B X. V))
4		sslin 2238	. . 3 |- ((A X. V) (_ (B X. V) -> (C i^i (A X. V)) (_ (C i^i (B X. V)))
5	3, 4	syl 10	. 2 |- (A (_ B -> (C i^i (A X. V)) (_ (C i^i (B X. V)))
6		df-res 3196	. 2 |- (C |` A) = (C i^i (A X. V))
7		df-res 3196	. 2 |- (C |` B) = (C i^i (B X. V))
8	5, 6, 7	3sstr4g 2105	1 |- (A (_ B -> (C |` A) (_ (C |` B))

Syntax hints:   -> wi 3  Vcvv 1814   i^i cin 2049   (_ wss 2050   X. cxp 3174   |` cres 3178

This theorem is referenced by:  imass2 3439  1stcof 4107

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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