Theorem algrflemg 6230

Description: Lemma for algrf 12039 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Jim Kingdon, 22-Jul-2021.)

Assertion

Ref	Expression
algrflemg	|- ( ( B e. V /\ C e. W ) -> ( B ( F o. 1st ) C ) = ( F ` B ) )

Proof of Theorem algrflemg

Step	Hyp	Ref	Expression
1		df-ov 5877	. 2 |- ( B ( F o. 1st ) C ) = ( ( F o. 1st ) ` <. B , C >. )
2		fo1st 6157	. . . . 5 |- 1st : _V -onto-> _V
3		fof 5438	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
4	2, 3	ax-mp 5	. . . 4 |- 1st : _V --> _V
5		opexg 4228	. . . 4 |- ( ( B e. V /\ C e. W ) -> <. B , C >. e. _V )
6		fvco3 5587	. . . 4 |- ( ( 1st : _V --> _V /\ <. B , C >. e. _V ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` ( 1st ` <. B , C >. ) ) )
7	4, 5, 6	sylancr 414	. . 3 |- ( ( B e. V /\ C e. W ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` ( 1st ` <. B , C >. ) ) )
8		op1stg 6150	. . . 4 |- ( ( B e. V /\ C e. W ) -> ( 1st ` <. B , C >. ) = B )
9	8	fveq2d 5519	. . 3 |- ( ( B e. V /\ C e. W ) -> ( F ` ( 1st ` <. B , C >. ) ) = ( F ` B ) )
10	7, 9	eqtrd 2210	. 2 |- ( ( B e. V /\ C e. W ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` B ) )
11	1, 10	eqtrid 2222	1 |- ( ( B e. V /\ C e. W ) -> ( B ( F o. 1st ) C ) = ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2737   <.cop 3595   o. ccom 4630   -->wf 5212   -onto->wfo 5214   ` cfv 5216  (class class class)co 5874   1stc1st 6138

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fo 5222  df-fv 5224  df-ov 5877  df-1st 6140

This theorem is referenced by:  ialgrlem1st  12036  algrp1  12040