Theorem algrflemg 6428

Description: Lemma for algrf 12746 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 6055	. 2 |- ( B ( F o. 1st ) C ) = ( ( F o. 1st ) ` <. B , C >. )
2		fo1st 6353	. . . . 5 |- 1st : _V -onto-> _V
3		fof 5592	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
4	2, 3	ax-mp 5	. . . 4 |- 1st : _V --> _V
5		opexg 4346	. . . 4 |- ( ( B e. V /\ C e. W ) -> <. B , C >. e. _V )
6		fvco3 5750	. . . 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 6346	. . . 4 |- ( ( B e. V /\ C e. W ) -> ( 1st ` <. B , C >. ) = B )
9	8	fveq2d 5676	. . 3 |- ( ( B e. V /\ C e. W ) -> ( F ` ( 1st ` <. B , C >. ) ) = ( F ` B ) )
10	7, 9	eqtrd 2267	. 2 |- ( ( B e. V /\ C e. W ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` B ) )
11	1, 10	eqtrid 2279	1 |- ( ( B e. V /\ C e. W ) -> ( B ( F o. 1st ) C ) = ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   <.cop 3694   o. ccom 4755   -->wf 5350   -onto->wfo 5352   ` cfv 5354  (class class class)co 6052   1stc1st 6334

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fo 5360  df-fv 5362  df-ov 6055  df-1st 6336

This theorem is referenced by:  ialgrlem1st  12743  algrp1  12747