Theorem algrflem 6282

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 5921	. 2 |- ( B ( F o. 1st ) C ) = ( ( F o. 1st ) ` <. B , C >. )
2		fo1st 6210	. . . 4 |- 1st : _V -onto-> _V
3		fof 5476	. . . 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 4257	. . . 4 |- ( ( B e. _V /\ C e. _V ) -> <. B , C >. e. _V )
8	5, 6, 7	mp2an 426	. . 3 |- <. B , C >. e. _V
9		fvco3 5628	. . 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 6199	. . 3 |- ( 1st ` <. B , C >. ) = B
12	11	fveq2i 5557	. 2 |- ( F ` ( 1st ` <. B , C >. ) ) = ( F ` B )
13	1, 10, 12	3eqtri 2218	1 |- ( B ( F o. 1st ) C ) = ( F ` B )

Syntax hints:   = wceq 1364   e. wcel 2164   _Vcvv 2760   <.cop 3621   o. ccom 4663   -->wf 5250   -onto->wfo 5252   ` cfv 5254  (class class class)co 5918   1stc1st 6191

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262  df-ov 5921  df-1st 6193

This theorem is referenced by:  algrf  12183