Theorem resabs1d 5428

Description: Absorption law for restriction, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
resabs1d.b	|- ( ph -> B C_ C )

Assertion

Ref	Expression
resabs1d	|- ( ph -> ( ( A |` C ) |` B ) = ( A |` B ) )

Proof of Theorem resabs1d

Step	Hyp	Ref	Expression
1		resabs1d.b	. 2 |- ( ph -> B C_ C )
2		resabs1 5427	. 2 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )
3	1, 2	syl 17	1 |- ( ph -> ( ( A |` C ) |` B ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1483   C_ wss 3574   |` cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121  df-res 5126

This theorem is referenced by:  f2ndf  7283  ablfac1eulem  18471  kgencn2  21360  tsmsres  21947  resubmet  22605  xrge0gsumle  22636  cmsss  23147  minveclem3a  23198  dvlip2  23758  c1liplem1  23759  efcvx  24203  logccv  24409  loglesqrt  24499  wilthlem2  24795  bnj1280  31088  cvmlift2lem9  31293  nosupno  31849  nosupbnd1lem1  31854  nosupbnd2  31862  mbfresfi  33456  ssbnd  33587  prdsbnd2  33594  cnpwstotbnd  33596  reheibor  33638  diophin  37336  fnwe2lem2  37621  dvsid  38530  limcresiooub  39874  limcresioolb  39875  dvmptresicc  40134  fourierdlem46  40369  fourierdlem48  40371  fourierdlem49  40372  fourierdlem58  40381  fourierdlem72  40395  fourierdlem73  40396  fourierdlem74  40397  fourierdlem75  40398  fourierdlem89  40412  fourierdlem90  40413  fourierdlem91  40414  fourierdlem93  40416  fourierdlem100  40423  fourierdlem102  40425  fourierdlem103  40426  fourierdlem104  40427  fourierdlem107  40430  fourierdlem111  40434  fourierdlem112  40435  fourierdlem114  40437  afvres  41252  funcrngcsetc  41998  funcrngcsetcALT  41999  funcringcsetc  42035