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Theorem algrflemg 5879
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 5543 . 2  |-  ( B ( F  o.  1st ) C )  =  ( ( F  o.  1st ) `  <. B ,  C >. )
2 fo1st 5812 . . . . 5  |-  1st : _V -onto-> _V
3 fof 5134 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
42, 3ax-mp 7 . . . 4  |-  1st : _V
--> _V
5 opexg 3992 . . . 4  |-  ( ( B  e.  V  /\  C  e.  W )  -> 
<. B ,  C >.  e. 
_V )
6 fvco3 5272 . . . 4  |-  ( ( 1st : _V --> _V  /\  <. B ,  C >.  e. 
_V )  ->  (
( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  ( 1st `  <. B ,  C >. ) ) )
74, 5, 6sylancr 399 . . 3  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( ( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  ( 1st `  <. B ,  C >. ) ) )
8 op1stg 5805 . . . 4  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( 1st `  <. B ,  C >. )  =  B )
98fveq2d 5210 . . 3  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( F `  ( 1st `  <. B ,  C >. ) )  =  ( F `  B ) )
107, 9eqtrd 2088 . 2  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( ( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  B ) )
111, 10syl5eq 2100 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 101    = wceq 1259    e. wcel 1409   _Vcvv 2574   <.cop 3406    o. ccom 4377   -->wf 4926   -onto->wfo 4928   ` cfv 4930  (class class class)co 5540   1stc1st 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-ov 5543  df-1st 5795
This theorem is referenced by:  ialgrlem1st  10264  ialgrp1  10268
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