Theorem caofid0l 6292

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 5564	. . 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 5508	. 2 |- ( ph -> F Fn A )
7		fvconst2g 5897	. . 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 2233	. 2 |- ( ( ph /\ w e. A ) -> ( F ` w ) = ( F ` w ) )
10		caofid0l.5	. . . 4 |- ( ( ph /\ x e. S ) -> ( B R x ) = x )
11	10	ralrimiva 2615	. . 3 |- ( ph -> A. x e. S ( B R x ) = x )
12	5	ffvelcdmda 5811	. . 3 |- ( ( ph /\ w e. A ) -> ( F ` w ) e. S )
13		oveq2 6057	. . . . 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 2247	. . . 4 |- ( x = ( F ` w ) -> ( ( B R x ) = x <-> ( B R ( F ` w ) ) = ( F ` w ) ) )
16	15	rspccva 2919	. . 3 |- ( ( A. x e. S ( B R x ) = x /\ ( F ` w ) e. S ) -> ( B R ( F ` w ) ) = ( F ` w ) )
17	11, 12, 16	syl2an2r 599	. 2 |- ( ( ph /\ w e. A ) -> ( B R ( F ` w ) ) = ( F ` w ) )
18	1, 4, 6, 6, 8, 9, 17	offveq 6286	1 |- ( ph -> ( ( A X. { B } ) oF R F ) = F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   A.wral 2520   {csn 3688   X. cxp 4746   Fn wfn 5346   -->wf 5347   ` cfv 5351  (class class class)co 6049   oFcof 6263

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-of 6265

This theorem is referenced by:  psr0lid  14829