Theorem ofc1g 6156

Description: Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
ofc1.1	|- ( ph -> A e. V )
ofc1.2	|- ( ph -> B e. W )
ofc1.3	|- ( ph -> F Fn A )
ofc1.4	|- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
ofc1g.ex	|- ( ( ph /\ X e. A ) -> ( B R C ) e. U )

Assertion

Ref	Expression
ofc1g	|- ( ( ph /\ X e. A ) -> ( ( ( A X. { B } ) oF R F ) ` X ) = ( B R C ) )

Proof of Theorem ofc1g

Step	Hyp	Ref	Expression
1		ofc1.2	. . 3 |- ( ph -> B e. W )
2		fnconstg 5455	. . 3 |- ( B e. W -> ( A X. { B } ) Fn A )
3	1, 2	syl 14	. 2 |- ( ph -> ( A X. { B } ) Fn A )
4		ofc1.3	. 2 |- ( ph -> F Fn A )
5		ofc1.1	. 2 |- ( ph -> A e. V )
6		inidm 3372	. 2 |- ( A i^i A ) = A
7		fvconst2g 5776	. . 3 |- ( ( B e. W /\ X e. A ) -> ( ( A X. { B } ) ` X ) = B )
8	1, 7	sylan 283	. 2 |- ( ( ph /\ X e. A ) -> ( ( A X. { B } ) ` X ) = B )
9		ofc1.4	. 2 |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
10		ofc1g.ex	. 2 |- ( ( ph /\ X e. A ) -> ( B R C ) e. U )
11	3, 4, 5, 5, 6, 8, 9, 10	ofvalg 6145	1 |- ( ( ph /\ X e. A ) -> ( ( ( A X. { B } ) oF R F ) ` X ) = ( B R C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   {csn 3622   X. cxp 4661   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   oFcof 6133

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-of 6135

This theorem is referenced by:  ofnegsub  8986