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Theorem algrflem 6892
Description: Lemma for algrf 14409 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 6280 . 2  |-  ( B ( F  o.  1st ) C )  =  ( ( F  o.  1st ) `  <. B ,  C >. )
2 fo1st 6803 . . . 4  |-  1st : _V -onto-> _V
3 fof 5777 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
42, 3ax-mp 5 . . 3  |-  1st : _V
--> _V
5 opex 4654 . . 3  |-  <. B ,  C >.  e.  _V
6 fvco3 5925 . . 3  |-  ( ( 1st : _V --> _V  /\  <. B ,  C >.  e. 
_V )  ->  (
( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  ( 1st `  <. B ,  C >. ) ) )
74, 5, 6mp2an 670 . 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 6791 . . 3  |-  ( 1st `  <. B ,  C >. )  =  B
1110fveq2i 5851 . 2  |-  ( F `
 ( 1st `  <. B ,  C >. )
)  =  ( F `
 B )
121, 7, 113eqtri 2435 1  |-  ( B ( F  o.  1st ) C )  =  ( F `  B )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405    e. wcel 1842   _Vcvv 3058   <.cop 3977    o. ccom 4826   -->wf 5564   -onto->wfo 5566   ` cfv 5568  (class class class)co 6277   1stc1st 6781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fo 5574  df-fv 5576  df-ov 6280  df-1st 6783
This theorem is referenced by:  fpwwe  9053  seq1st  14407  algrf  14409  algrp1  14410  dvnff  22616  dvnp1  22618
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