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Theorem algrflemg 5977
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
StepHypRef Expression
1 df-ov 5637 . 2  |-  ( B ( F  o.  1st ) C )  =  ( ( F  o.  1st ) `  <. B ,  C >. )
2 fo1st 5910 . . . . 5  |-  1st : _V -onto-> _V
3 fof 5217 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
42, 3ax-mp 7 . . . 4  |-  1st : _V
--> _V
5 opexg 4046 . . . 4  |-  ( ( B  e.  V  /\  C  e.  W )  -> 
<. B ,  C >.  e. 
_V )
6 fvco3 5359 . . . 4  |-  ( ( 1st : _V --> _V  /\  <. B ,  C >.  e. 
_V )  ->  (
( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  ( 1st `  <. B ,  C >. ) ) )
74, 5, 6sylancr 405 . . 3  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( ( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  ( 1st `  <. B ,  C >. ) ) )
8 op1stg 5903 . . . 4  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( 1st `  <. B ,  C >. )  =  B )
98fveq2d 5293 . . 3  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( F `  ( 1st `  <. B ,  C >. ) )  =  ( F `  B ) )
107, 9eqtrd 2120 . 2  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( ( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  B ) )
111, 10syl5eq 2132 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 102    = wceq 1289    e. wcel 1438   _Vcvv 2619   <.cop 3444    o. ccom 4432   -->wf 4998   -onto->wfo 5000   ` cfv 5002  (class class class)co 5634   1stc1st 5891
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fo 5008  df-fv 5010  df-ov 5637  df-1st 5893
This theorem is referenced by:  ialgrlem1st  11106  ialgrp1  11110
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