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Theorem algrflem 6315
Description: Lemma for algrf 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 5947 . 2  |-  ( B ( F  o.  1st ) C )  =  ( ( F  o.  1st ) `  <. B ,  C >. )
2 fo1st 6243 . . . 4  |-  1st : _V -onto-> _V
3 fof 5498 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st
: _V --> _V )
42, 3ax-mp 5 . . 3  |-  1st : _V
--> _V
5 algrflem.1 . . . 4  |-  B  e. 
_V
6 algrflem.2 . . . 4  |-  C  e. 
_V
7 opexg 4272 . . . 4  |-  ( ( B  e.  _V  /\  C  e.  _V )  -> 
<. B ,  C >.  e. 
_V )
85, 6, 7mp2an 426 . . 3  |-  <. B ,  C >.  e.  _V
9 fvco3 5650 . . 3  |-  ( ( 1st : _V --> _V  /\  <. B ,  C >.  e. 
_V )  ->  (
( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  ( 1st `  <. B ,  C >. ) ) )
104, 8, 9mp2an 426 . 2  |-  ( ( F  o.  1st ) `  <. B ,  C >. )  =  ( F `
 ( 1st `  <. B ,  C >. )
)
115, 6op1st 6232 . . 3  |-  ( 1st `  <. B ,  C >. )  =  B
1211fveq2i 5579 . 2  |-  ( F `
 ( 1st `  <. B ,  C >. )
)  =  ( F `
 B )
131, 10, 123eqtri 2230 1  |-  ( B ( F  o.  1st ) C )  =  ( F `  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1373    e. wcel 2176   _Vcvv 2772   <.cop 3636    o. ccom 4679   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5944   1stc1st 6224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-ov 5947  df-1st 6226
This theorem is referenced by:  algrf  12367
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