Theorem algrflem 6373

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 6003	. 2 |- ( B ( F o. 1st ) C ) = ( ( F o. 1st ) ` <. B , C >. )
2		fo1st 6301	. . . 4 |- 1st : _V -onto-> _V
3		fof 5547	. . . 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 4313	. . . 4 |- ( ( B e. _V /\ C e. _V ) -> <. B , C >. e. _V )
8	5, 6, 7	mp2an 426	. . 3 |- <. B , C >. e. _V
9		fvco3 5704	. . 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 6290	. . 3 |- ( 1st ` <. B , C >. ) = B
12	11	fveq2i 5629	. 2 |- ( F ` ( 1st ` <. B , C >. ) ) = ( F ` B )
13	1, 10, 12	3eqtri 2254	1 |- ( B ( F o. 1st ) C ) = ( F ` B )

Syntax hints:   = wceq 1395   e. wcel 2200   _Vcvv 2799   <.cop 3669   o. ccom 4722   -->wf 5313   -onto->wfo 5315   ` cfv 5317  (class class class)co 6000   1stc1st 6282

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fo 5323  df-fv 5325  df-ov 6003  df-1st 6284

This theorem is referenced by:  algrf  12562