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Theorem algrflem 6224
Description: Lemma for algrf 12743 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 5861 . 2  |-  ( B ( F  o.  1st ) C )  =  ( ( F  o.  1st ) `  <. B ,  C >. )
2 fo1st 6139 . . . 4  |-  1st : _V -onto-> _V
3 fof 5451 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
42, 3ax-mp 8 . . 3  |-  1st : _V
--> _V
5 opex 4237 . . 3  |-  <. B ,  C >.  e.  _V
6 fvco3 5596 . . 3  |-  ( ( 1st : _V --> _V  /\  <. B ,  C >.  e. 
_V )  ->  (
( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  ( 1st `  <. B ,  C >. ) ) )
74, 5, 6mp2an 653 . 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 6128 . . 3  |-  ( 1st `  <. B ,  C >. )  =  B
1110fveq2i 5528 . 2  |-  ( F `
 ( 1st `  <. B ,  C >. )
)  =  ( F `
 B )
121, 7, 113eqtri 2307 1  |-  ( B ( F  o.  1st ) C )  =  ( F `  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    o. ccom 4693   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   1stc1st 6120
This theorem is referenced by:  fpwwe  8268  seq1st  12741  algrf  12743  algrp1  12744  dvnff  19272  dvnp1  19274
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122
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