Theorem algrflemg 6135

Description: Lemma for algrf 11762 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 5785	. 2 |- ( B ( F o. 1st ) C ) = ( ( F o. 1st ) ` <. B , C >. )
2		fo1st 6063	. . . . 5 |- 1st : _V -onto-> _V
3		fof 5353	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
4	2, 3	ax-mp 5	. . . 4 |- 1st : _V --> _V
5		opexg 4158	. . . 4 |- ( ( B e. V /\ C e. W ) -> <. B , C >. e. _V )
6		fvco3 5500	. . . 4 |- ( ( 1st : _V --> _V /\ <. B , C >. e. _V ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` ( 1st ` <. B , C >. ) ) )
7	4, 5, 6	sylancr 411	. . 3 |- ( ( B e. V /\ C e. W ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` ( 1st ` <. B , C >. ) ) )
8		op1stg 6056	. . . 4 |- ( ( B e. V /\ C e. W ) -> ( 1st ` <. B , C >. ) = B )
9	8	fveq2d 5433	. . 3 |- ( ( B e. V /\ C e. W ) -> ( F ` ( 1st ` <. B , C >. ) ) = ( F ` B ) )
10	7, 9	eqtrd 2173	. 2 |- ( ( B e. V /\ C e. W ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` B ) )
11	1, 10	syl5eq 2185	1 |- ( ( B e. V /\ C e. W ) -> ( B ( F o. 1st ) C ) = ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   _Vcvv 2689   <.cop 3535   o. ccom 4551   -->wf 5127   -onto->wfo 5129   ` cfv 5131  (class class class)co 5782   1stc1st 6044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fo 5137  df-fv 5139  df-ov 5785  df-1st 6046

This theorem is referenced by:  ialgrlem1st  11759  algrp1  11763