Theorem ofmresval 5723

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 5640	. 2 |- ( ( F e. A /\ G e. B ) -> ( F ( oF R |` ( A X. B ) ) G ) = ( F oF R G ) )
4	1, 2, 3	syl2anc 391	1 |- ( ph -> ( F ( oF R |` ( A X. B ) ) G ) = ( F oF R G ) )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   X. cxp 4343   |` cres 4347  (class class class)co 5512   oFcof 5710

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-res 4357  df-iota 4867  df-fv 4910  df-ov 5515

This theorem is referenced by: (None)