Theorem algrflem 6393

Description: Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
algrflem.1	|- B e. _V
algrflem.2	|- C e. _V

Assertion

Ref	Expression
algrflem	|- ( B ( F o. 1st ) C ) = ( F ` B )

Proof of Theorem algrflem

Step	Hyp	Ref	Expression
1		df-ov 6020	. 2 |- ( B ( F o. 1st ) C ) = ( ( F o. 1st ) ` <. B , C >. )
2		fo1st 6319	. . . 4 |- 1st : _V -onto-> _V
3		fof 5559	. . . 4 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
4	2, 3	ax-mp 5	. . 3 |- 1st : _V --> _V
5		algrflem.1	. . . 4 |- B e. _V
6		algrflem.2	. . . 4 |- C e. _V
7		opexg 4320	. . . 4 |- ( ( B e. _V /\ C e. _V ) -> <. B , C >. e. _V )
8	5, 6, 7	mp2an 426	. . 3 |- <. B , C >. e. _V
9		fvco3 5717	. . 3 |- ( ( 1st : _V --> _V /\ <. B , C >. e. _V ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` ( 1st ` <. B , C >. ) ) )
10	4, 8, 9	mp2an 426	. 2 |- ( ( F o. 1st ) ` <. B , C >. ) = ( F ` ( 1st ` <. B , C >. ) )
11	5, 6	op1st 6308	. . 3 |- ( 1st ` <. B , C >. ) = B
12	11	fveq2i 5642	. 2 |- ( F ` ( 1st ` <. B , C >. ) ) = ( F ` B )
13	1, 10, 12	3eqtri 2256	1 |- ( B ( F o. 1st ) C ) = ( F ` B )

Syntax hints:   = wceq 1397   e. wcel 2202   _Vcvv 2802   <.cop 3672   o. ccom 4729   -->wf 5322   -onto->wfo 5324   ` cfv 5326  (class class class)co 6017   1stc1st 6300

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-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-ov 6020  df-1st 6302

This theorem is referenced by:  algrf  12616