Theorem ofmresval 6138

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 6054	. 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 1364   e. wcel 2164   X. cxp 4655   |` cres 4659  (class class class)co 5914   oFcof 6124

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4663  df-res 4669  df-iota 5211  df-fv 5258  df-ov 5917

This theorem is referenced by:  psradd  14144