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Theorem algrflemg 6095
Description: Lemma for algrf 11653 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 5745 . 2  |-  ( B ( F  o.  1st ) C )  =  ( ( F  o.  1st ) `  <. B ,  C >. )
2 fo1st 6023 . . . . 5  |-  1st : _V -onto-> _V
3 fof 5315 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
42, 3ax-mp 5 . . . 4  |-  1st : _V
--> _V
5 opexg 4120 . . . 4  |-  ( ( B  e.  V  /\  C  e.  W )  -> 
<. B ,  C >.  e. 
_V )
6 fvco3 5460 . . . 4  |-  ( ( 1st : _V --> _V  /\  <. B ,  C >.  e. 
_V )  ->  (
( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  ( 1st `  <. B ,  C >. ) ) )
74, 5, 6sylancr 410 . . 3  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( ( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  ( 1st `  <. B ,  C >. ) ) )
8 op1stg 6016 . . . 4  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( 1st `  <. B ,  C >. )  =  B )
98fveq2d 5393 . . 3  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( F `  ( 1st `  <. B ,  C >. ) )  =  ( F `  B ) )
107, 9eqtrd 2150 . 2  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( ( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  B ) )
111, 10syl5eq 2162 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 103    = wceq 1316    e. wcel 1465   _Vcvv 2660   <.cop 3500    o. ccom 4513   -->wf 5089   -onto->wfo 5091   ` cfv 5093  (class class class)co 5742   1stc1st 6004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-ov 5745  df-1st 6006
This theorem is referenced by:  ialgrlem1st  11650  algrp1  11654
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