Theorem ofc1g 6324

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 5590	. . 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 3440	. 2 |- ( A i^i A ) = A
7		fvconst2g 5929	. . 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 6312	1 |- ( ( ph /\ X e. A ) -> ( ( ( A X. { B } ) oF R F ) ` X ) = ( B R C ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {csn 3709   X. cxp 4772   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   oFcof 6300

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302

This theorem is used by:  ofnegsub  9293