Theorem algrflemg 6316

Description: Lemma for algrf 12367 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 5947	. 2 |- ( B ( F o. 1st ) C ) = ( ( F o. 1st ) ` <. B , C >. )
2		fo1st 6243	. . . . 5 |- 1st : _V -onto-> _V
3		fof 5498	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
4	2, 3	ax-mp 5	. . . 4 |- 1st : _V --> _V
5		opexg 4272	. . . 4 |- ( ( B e. V /\ C e. W ) -> <. B , C >. e. _V )
6		fvco3 5650	. . . 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 6236	. . . 4 |- ( ( B e. V /\ C e. W ) -> ( 1st ` <. B , C >. ) = B )
9	8	fveq2d 5580	. . 3 |- ( ( B e. V /\ C e. W ) -> ( F ` ( 1st ` <. B , C >. ) ) = ( F ` B ) )
10	7, 9	eqtrd 2238	. 2 |- ( ( B e. V /\ C e. W ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` B ) )
11	1, 10	eqtrid 2250	1 |- ( ( B e. V /\ C e. W ) -> ( B ( F o. 1st ) C ) = ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   _Vcvv 2772   <.cop 3636   o. ccom 4679   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5944   1stc1st 6224

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5947  df-1st 6226

This theorem is referenced by:  ialgrlem1st  12364  algrp1  12368