Theorem resabs1d 4921

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 4920	. 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 1348   C_ wss 3121   |` cres 4613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617  df-rel 4618  df-res 4623

This theorem is referenced by:  resubmet  13342