Theorem algrflem 6425

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 6053	. 2 |- ( B ( F o. 1st ) C ) = ( ( F o. 1st ) ` <. B , C >. )
2		fo1st 6351	. . . 4 |- 1st : _V -onto-> _V
3		fof 5590	. . . 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 4344	. . . 4 |- ( ( B e. _V /\ C e. _V ) -> <. B , C >. e. _V )
8	5, 6, 7	mp2an 426	. . 3 |- <. B , C >. e. _V
9		fvco3 5748	. . 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 6340	. . 3 |- ( 1st ` <. B , C >. ) = B
12	11	fveq2i 5673	. 2 |- ( F ` ( 1st ` <. B , C >. ) ) = ( F ` B )
13	1, 10, 12	3eqtri 2257	1 |- ( B ( F o. 1st ) C ) = ( F ` B )

Syntax hints:   = wceq 1398   e. wcel 2203   _Vcvv 2813   <.cop 3692   o. ccom 4753   -->wf 5348   -onto->wfo 5350   ` cfv 5352  (class class class)co 6050   1stc1st 6332

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 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-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  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-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-ov 6053  df-1st 6334

This theorem is referenced by:  algrf  12742