Theorem ofc1g 6151

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 5451	. . 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 3368	. 2 |- ( A i^i A ) = A
7		fvconst2g 5772	. . 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 6140	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 2164   {csn 3618   X. cxp 4657   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   oFcof 6128

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 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-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  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-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130

This theorem is referenced by:  ofnegsub  8981