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Theorem resabs1 4583
Description: Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
Assertion
Ref Expression
resabs1 
C_  C  |`  C  |`  |`

Proof of Theorem resabs1
StepHypRef Expression
1 resres 4567 . 2  |`  C  |`  |`  C  i^i
2 sseqin2 3150 . . 3 
C_  C  C  i^i
3 reseq2 4550 . . 3  C  i^i  |`  C  i^i  |`
42, 3sylbi 114 . 2 
C_  C  |`  C  i^i  |`
51, 4syl5eq 2081 1 
C_  C  |`  C  |`  |`
Colors of variables: wff set class
Syntax hints:   wi 4   wceq 1242    i^i cin 2910    C_ wss 2911    |` cres 4290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294  df-rel 4295  df-res 4300
This theorem is referenced by:  resabs2  4584  resiima  4626  fun2ssres  4886  fssres2  5010  f2ndf  5789  smores3  5849  tfrlemisucaccv  5880
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