Theorem ofc2g 6258

Description: Right operation by a constant. (Contributed by NM, 7-Oct-2014.)

Hypotheses

Ref	Expression
ofc2.1	|- ( ph -> A e. V )
ofc2.2	|- ( ph -> B e. W )
ofc2.3	|- ( ph -> F Fn A )
ofc2.4	|- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
ofc2g.ex	|- ( ( ph /\ X e. A ) -> ( C R B ) e. U )

Assertion

Ref	Expression
ofc2g	|- ( ( ph /\ X e. A ) -> ( ( F oF R ( A X. { B } ) ) ` X ) = ( C R B ) )

Proof of Theorem ofc2g

Step	Hyp	Ref	Expression
1		ofc2.3	. 2 |- ( ph -> F Fn A )
2		ofc2.2	. . 3 |- ( ph -> B e. W )
3		fnconstg 5534	. . 3 |- ( B e. W -> ( A X. { B } ) Fn A )
4	2, 3	syl 14	. 2 |- ( ph -> ( A X. { B } ) Fn A )
5		ofc2.1	. 2 |- ( ph -> A e. V )
6		inidm 3416	. 2 |- ( A i^i A ) = A
7		ofc2.4	. 2 |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
8		fvconst2g 5868	. . 3 |- ( ( B e. W /\ X e. A ) -> ( ( A X. { B } ) ` X ) = B )
9	2, 8	sylan 283	. 2 |- ( ( ph /\ X e. A ) -> ( ( A X. { B } ) ` X ) = B )
10		ofc2g.ex	. 2 |- ( ( ph /\ X e. A ) -> ( C R B ) e. U )
11	1, 4, 5, 5, 6, 7, 9, 10	ofvalg 6245	1 |- ( ( ph /\ X e. A ) -> ( ( F oF R ( A X. { B } ) ) ` X ) = ( C R B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   {csn 3669   X. cxp 4723   Fn wfn 5321   ` cfv 5326  (class class class)co 6018   oFcof 6233

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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235

This theorem is referenced by: (None)