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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
StepHypRef 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 )
42, 3ax-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 >. ) ) )
74, 5, 6sylancr 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 )
98fveq2d 5519 . . 3  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( F `  ( 1st `  <. B ,  C >. ) )  =  ( F `  B ) )
107, 9eqtrd 2210 . 2  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( ( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  B ) )
111, 10eqtrid 2222 1  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( B ( F  o.  1st ) C )  =  ( F `
 B ) )
Colors of variables: wff set class
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
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