Theorem ofc1g 6192

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 5484	. . 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 3386	. 2 |- ( A i^i A ) = A
7		fvconst2g 5810	. . 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 6180	1 |- ( ( ph /\ X e. A ) -> ( ( ( A X. { B } ) oF R F ) ` X ) = ( B R C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   {csn 3637   X. cxp 4680   Fn wfn 5274   ` cfv 5279  (class class class)co 5956   oFcof 6168

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-pow 4225  ax-pr 4260  ax-setind 4592

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-ov 5959  df-oprab 5960  df-mpo 5961  df-of 6170

This theorem is referenced by:  ofnegsub  9050