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Theorem algrflem 6092
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 5743 . 2  |-  ( B ( F  o.  1st ) C )  =  ( ( F  o.  1st ) `  <. B ,  C >. )
2 fo1st 6021 . . . 4  |-  1st : _V -onto-> _V
3 fof 5313 . . . 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 4118 . . . 4  |-  ( ( B  e.  _V  /\  C  e.  _V )  -> 
<. B ,  C >.  e. 
_V )
85, 6, 7mp2an 420 . . 3  |-  <. B ,  C >.  e.  _V
9 fvco3 5458 . . 3  |-  ( ( 1st : _V --> _V  /\  <. B ,  C >.  e. 
_V )  ->  (
( F  o.  1st ) `  <. B ,  C >. )  =  ( F `  ( 1st `  <. B ,  C >. ) ) )
104, 8, 9mp2an 420 . 2  |-  ( ( F  o.  1st ) `  <. B ,  C >. )  =  ( F `
 ( 1st `  <. B ,  C >. )
)
115, 6op1st 6010 . . 3  |-  ( 1st `  <. B ,  C >. )  =  B
1211fveq2i 5390 . 2  |-  ( F `
 ( 1st `  <. B ,  C >. )
)  =  ( F `
 B )
131, 10, 123eqtri 2140 1  |-  ( B ( F  o.  1st ) C )  =  ( F `  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1314    e. wcel 1463   _Vcvv 2658   <.cop 3498    o. ccom 4511   -->wf 5087   -onto->wfo 5089   ` cfv 5091  (class class class)co 5740   1stc1st 6002
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fo 5097  df-fv 5099  df-ov 5743  df-1st 6004
This theorem is referenced by:  algrf  11622
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