Theorem xpssres 5434

Description: Restriction of a constant function (or other Cartesian product). (Contributed by Stefan O'Rear, 24-Jan-2015.)

Assertion

Ref	Expression
xpssres	|- ( C C_ A -> ( ( A X. B ) |` C ) = ( C X. B ) )

Proof of Theorem xpssres

Step	Hyp	Ref	Expression
1		df-res 5126	. . 3 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
2		inxp 5254	. . 3 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
3		inv1 3970	. . . 4 |- ( B i^i _V ) = B
4	3	xpeq2i 5136	. . 3 |- ( ( A i^i C ) X. ( B i^i _V ) ) = ( ( A i^i C ) X. B )
5	1, 2, 4	3eqtri 2648	. 2 |- ( ( A X. B ) |` C ) = ( ( A i^i C ) X. B )
6		sseqin2 3817	. . . 4 |- ( C C_ A <-> ( A i^i C ) = C )
7	6	biimpi 206	. . 3 |- ( C C_ A -> ( A i^i C ) = C )
8	7	xpeq1d 5138	. 2 |- ( C C_ A -> ( ( A i^i C ) X. B ) = ( C X. B ) )
9	5, 8	syl5eq 2668	1 |- ( C C_ A -> ( ( A X. B ) |` C ) = ( C X. B ) )

Syntax hints:   -> wi 4   = wceq 1483   _Vcvv 3200   i^i cin 3573   C_ wss 3574   X. cxp 5112   |` 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:  fparlem3  7279  fparlem4  7280  fpwwe2lem13  9464  pwssplit3  19061  cnconst2  21087  xkoccn  21422  tmdgsum  21899  dvcmul  23707  dvcmulf  23708  dvsconst  38529  dvsid  38530  aacllem  42547