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Theorem domcatval2 26034
Description: The domain of a morphism in the category Set. (Contributed by FL, 6-Nov-2013.)
Hypotheses
Ref Expression
domcatval2.1  |- .Morphism  =  ( Morphism SetCat `  U )
domcatval2.2  |- .dom  =  ( dom SetCat `  U
)
Assertion
Ref Expression
domcatval2  |-  ( ( U  e.  Univ  /\  A  e. .Morphism  )  ->  (.dom  `  A )  =  ( ( 1st  o.  1st ) `  A )
)

Proof of Theorem domcatval2
StepHypRef Expression
1 domcatval2.1 . 2  |- .Morphism  =  ( Morphism SetCat `  U )
2 eleq2 2357 . . . 4  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( A  e. .Morphism  <->  A  e.  ( Morphism SetCat `  U )
) )
32anbi2d 684 . . 3  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( ( U  e. 
Univ  /\  A  e. .Morphism  ) 
<->  ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U ) ) ) )
4 domcatval2.2 . . . 4  |- .dom  =  ( dom SetCat `  U
)
5 domcatval 26033 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  ( ( dom
SetCat `  U ) `  A )  =  ( ( 1st  o.  1st ) `  A )
)
6 fveq1 5540 . . . . . 6  |-  (.dom  =  ( dom SetCat `  U
)  ->  (.dom  `  A )  =  ( ( dom SetCat `  U
) `  A )
)
76eqeq1d 2304 . . . . 5  |-  (.dom  =  ( dom SetCat `  U
)  ->  ( (.dom  `  A )  =  ( ( 1st  o.  1st ) `  A )  <->  ( ( dom SetCat `  U
) `  A )  =  ( ( 1st 
o.  1st ) `  A
) ) )
85, 7syl5ibr 212 . . . 4  |-  (.dom  =  ( dom SetCat `  U
)  ->  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  (.dom  `  A )  =  ( ( 1st  o.  1st ) `  A )
) )
94, 8ax-mp 8 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  ( Morphism SetCat `  U )
)  ->  (.dom  `  A )  =  ( ( 1st  o.  1st ) `  A )
)
103, 9syl6bi 219 . 2  |-  (.Morphism  =  ( Morphism SetCat `  U )  ->  ( ( U  e. 
Univ  /\  A  e. .Morphism  )  ->  (.dom  `  A )  =  ( ( 1st  o.  1st ) `  A )
) )
111, 10ax-mp 8 1  |-  ( ( U  e.  Univ  /\  A  e. .Morphism  )  ->  (.dom  `  A )  =  ( ( 1st  o.  1st ) `  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    o. ccom 4709   ` cfv 5271   1stc1st 6136   Univcgru 8428   Morphism SetCatccmrcase 26013   dom
SetCatcdomcase 26022
This theorem is referenced by:  morexcmp  26070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-domcatset 26023
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