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Theorem algrflem 6882
Description: Lemma for algrf 14050 and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
algrflem.1  |-  B  e. 
_V
algrflem.2  |-  C  e. 
_V
Assertion
Ref Expression
algrflem  |-  ( B ( F  o.  1st ) C )  =  ( F `  B )

Proof of Theorem algrflem
StepHypRef Expression
1 df-ov 6278 . 2  |-  ( B ( F  o.  1st ) C )  =  ( ( F  o.  1st ) `  <. B ,  C >. )
2 fo1st 6794 . . . 4  |-  1st : _V -onto-> _V
3 fof 5786 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
42, 3ax-mp 5 . . 3  |-  1st : _V
--> _V
5 opex 4704 . . 3  |-  <. B ,  C >.  e.  _V
6 fvco3 5935 . . 3  |-  ( ( 1st : _V --> _V  /\  <. B ,  C >.  e. 
_V )  ->  (
( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  ( 1st `  <. B ,  C >. ) ) )
74, 5, 6mp2an 672 . 2  |-  ( ( F  o.  1st ) `  <. B ,  C >. )  =  ( F `
 ( 1st `  <. B ,  C >. )
)
8 algrflem.1 . . . 4  |-  B  e. 
_V
9 algrflem.2 . . . 4  |-  C  e. 
_V
108, 9op1st 6782 . . 3  |-  ( 1st `  <. B ,  C >. )  =  B
1110fveq2i 5860 . 2  |-  ( F `
 ( 1st `  <. B ,  C >. )
)  =  ( F `
 B )
121, 7, 113eqtri 2493 1  |-  ( B ( F  o.  1st ) C )  =  ( F `  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762   _Vcvv 3106   <.cop 4026    o. ccom 4996   -->wf 5575   -onto->wfo 5577   ` cfv 5579  (class class class)co 6275   1stc1st 6772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-ov 6278  df-1st 6774
This theorem is referenced by:  fpwwe  9013  seq1st  14048  algrf  14050  algrp1  14051  dvnff  22054  dvnp1  22056
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