Theorem ofmresval 6170

Description: Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)

Hypotheses

Ref	Expression
ofmresval.f	|- ( ph -> F e. A )
ofmresval.g	|- ( ph -> G e. B )

Assertion

Ref	Expression
ofmresval	|- ( ph -> ( F ( oF R |` ( A X. B ) ) G ) = ( F oF R G ) )

Proof of Theorem ofmresval

Step	Hyp	Ref	Expression
1		ofmresval.f	. 2 |- ( ph -> F e. A )
2		ofmresval.g	. 2 |- ( ph -> G e. B )
3		ovres 6086	. 2 |- ( ( F e. A /\ G e. B ) -> ( F ( oF R |` ( A X. B ) ) G ) = ( F oF R G ) )
4	1, 2, 3	syl2anc 411	1 |- ( ph -> ( F ( oF R |` ( A X. B ) ) G ) = ( F oF R G ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   X. cxp 4673   |` cres 4677  (class class class)co 5944   oFcof 6156

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-res 4687  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  psradd  14441