Theorem caofid0l 6323

Description: Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)

Hypotheses

Ref	Expression
caofref.1	|- ( ph -> A e. V )
caofref.2	|- ( ph -> F : A --> S )
caofid0.3	|- ( ph -> B e. W )
caofid0l.5	|- ( ( ph /\ x e. S ) -> ( B R x ) = x )

Assertion

Ref	Expression
caofid0l	|- ( ph -> ( ( A X. { B } ) oF R F ) = F )

Distinct variable groups:   x, B   x, F   ph, x   x, R   x, S

Allowed substitution hints:   A( x)   V( x)   W( x)

Proof of Theorem caofid0l

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caofref.1	. 2 |- ( ph -> A e. V )
2		caofid0.3	. . 3 |- ( ph -> B e. W )
3		fnconstg 5588	. . 3 |- ( B e. W -> ( A X. { B } ) Fn A )
4	2, 3	syl 14	. 2 |- ( ph -> ( A X. { B } ) Fn A )
5		caofref.2	. . 3 |- ( ph -> F : A --> S )
6	5	ffnd 5532	. 2 |- ( ph -> F Fn A )
7		fvconst2g 5923	. . 3 |- ( ( B e. W /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
8	2, 7	sylan 283	. 2 |- ( ( ph /\ w e. A ) -> ( ( A X. { B } ) ` w ) = B )
9		eqidd 2239	. 2 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
10		caofid0l.5	. . . 4 |- ( ( ph /\ x e. S ) -> ( B R x ) = x )
11	10	ralrimiva 2623	. . 3 |- ( ph -> A. x e. S ( B R x ) = x )
12	5	ffvelcdmda 5837	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
13		oveq2 6087	. . . . 5 |- ( x = ( F ` w ) -> ( B R x ) = ( B R ( F ` w ) ) )
14		id 19	. . . . 5 |- ( x = ( F ` w ) -> x = ( F ` w ) )
15	13, 14	eqeq12d 2253	. . . 4 |- ( x = ( F ` w ) -> ( ( B R x ) = x <-> ( B R ( F ` w ) ) = ( F ` w ) ) )
16	15	rspccva 2928	. . 3 |- ( ( A. x e. S ( B R x ) = x /\ ( F ` w ) e. S ) -> ( B R ( F ` w ) ) = ( F ` w ) )
17	11, 12, 16	syl2an2r 603	. 2 |- ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = ( F ` w ) )
18	1, 4, 6, 6, 8, 9, 17	offveq 6317	1 |- ( ph -> ( ( A X. { B } ) oF R F ) = F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   {csn 3708   X. cxp 4770   Fn wfn 5370   -->wf 5371   ` cfv 5375  (class class class)co 6079   oFcof 6294

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 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 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-setind 4682

This theorem 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 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-of 6296

This theorem is referenced by:  psr0lid  15056