Theorem algrflemg 5871

Description: Lemma for algrf and related theorems. (Contributed 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 5535	. 2 |- ( B ( F o. 1st ) C ) = ( ( F o. 1st ) ` <. B , C >. )
2		fo1st 5804	. . . . 5 |- 1st : _V -onto-> _V
3		fof 5126	. . . . 5 |- ( 1st : _V -onto-> _V -> 1st : _V --> _V )
4	2, 3	ax-mp 7	. . . 4 |- 1st : _V --> _V
5		opexg 3983	. . . 4 |- ( ( B e. V /\ C e. W ) -> <. B , C >. e. _V )
6		fvco3 5265	. . . 4 |- ( ( 1st : _V --> _V /\ <. B , C >. e. _V ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` ( 1st ` <. B , C >. ) ) )
7	4, 5, 6	sylancr 405	. . 3 |- ( ( B e. V /\ C e. W ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` ( 1st ` <. B , C >. ) ) )
8		op1stg 5797	. . . 4 |- ( ( B e. V /\ C e. W ) -> ( 1st ` <. B , C >. ) = B )
9	8	fveq2d 5202	. . 3 |- ( ( B e. V /\ C e. W ) -> ( F ` ( 1st ` <. B , C >. ) ) = ( F ` B ) )
10	7, 9	eqtrd 2113	. 2 |- ( ( B e. V /\ C e. W ) -> ( ( F o. 1st ) ` <. B , C >. ) = ( F ` B ) )
11	1, 10	syl5eq 2125	1 |- ( ( B e. V /\ C e. W ) -> ( B ( F o. 1st ) C ) = ( F ` B ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   <.cop 3401   o. ccom 4367   -->wf 4918   -onto->wfo 4920   ` cfv 4922  (class class class)co 5532   1stc1st 5785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930  df-ov 5535  df-1st 5787

This theorem is referenced by:  ialgrlem1st  10424  ialgrp1  10428