Theorem resabs1d 4988

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 4987	. 2 |- ( B C_ C -> ( ( A |` C ) |` B ) = ( A |` B ) )
3	1, 2	syl 14	1 |- ( ph -> ( ( A |` C ) |` B ) = ( A |` B ) )

Syntax hints:   -> wi 4   = wceq 1372   C_ wss 3165   |` cres 4676

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-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-opab 4105  df-xp 4680  df-rel 4681  df-res 4686

This theorem is referenced by:  resubmet  14999