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Theorem grphidmor2 25953
Description: Graph of an identity morphism. (Contributed by FL, 6-Nov-2013.)
Hypotheses
Ref Expression
grphidmor2.1  |- .graph  =  2nd
grphidmor2.2  |- .id  =  ( Id SetCat `  U
)
Assertion
Ref Expression
grphidmor2  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (.graph  `  (.id  `  A
) )  =  (  _I  |`  A )
)

Proof of Theorem grphidmor2
StepHypRef Expression
1 grphidmor2.2 . 2  |- .id  =  ( Id SetCat `  U
)
2 grphidmor2.1 . . . 4  |- .graph  =  2nd
3 idmor 25946 . . . . . 6  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( Id SetCat `  U
) `  A )  =  <. <. A ,  A >. ,  (  _I  |`  A )
>. )
4 fveq2 5525 . . . . . . 7  |-  ( ( ( Id SetCat `  U
) `  A )  =  <. <. A ,  A >. ,  (  _I  |`  A )
>.  ->  ( 2nd `  (
( Id SetCat `  U
) `  A )
)  =  ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A )
>. ) )
5 opex 4237 . . . . . . . . 9  |-  <. A ,  A >.  e.  _V
6 resiexg 4997 . . . . . . . . . 10  |-  ( A  e.  U  ->  (  _I  |`  A )  e. 
_V )
76adantl 452 . . . . . . . . 9  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (  _I  |`  A )  e. 
_V )
8 op2ndg 6133 . . . . . . . . 9  |-  ( (
<. A ,  A >.  e. 
_V  /\  (  _I  |`  A )  e.  _V )  ->  ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  =  (  _I  |`  A ) )
95, 7, 8sylancr 644 . . . . . . . 8  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  =  (  _I  |`  A ) )
10 eqtr 2300 . . . . . . . . 9  |-  ( ( ( 2nd `  (
( Id SetCat `  U
) `  A )
)  =  ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  /\  ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  =  (  _I  |`  A ) )  ->  ( 2nd `  (
( Id SetCat `  U
) `  A )
)  =  (  _I  |`  A ) )
1110expcom 424 . . . . . . . 8  |-  ( ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A ) >. )  =  (  _I  |`  A )  ->  ( ( 2nd `  ( ( Id SetCat `  U ) `  A
) )  =  ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A ) >. )  ->  ( 2nd `  (
( Id SetCat `  U
) `  A )
)  =  (  _I  |`  A ) ) )
129, 11syl 15 . . . . . . 7  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (
( 2nd `  (
( Id SetCat `  U
) `  A )
)  =  ( 2nd `  <. <. A ,  A >. ,  (  _I  |`  A )
>. )  ->  ( 2nd `  ( ( Id SetCat `  U ) `  A
) )  =  (  _I  |`  A )
) )
134, 12syl5com 26 . . . . . 6  |-  ( ( ( Id SetCat `  U
) `  A )  =  <. <. A ,  A >. ,  (  _I  |`  A )
>.  ->  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( 2nd `  ( ( Id SetCat `
 U ) `  A ) )  =  (  _I  |`  A ) ) )
143, 13mpcom 32 . . . . 5  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  ( 2nd `  ( ( Id SetCat `
 U ) `  A ) )  =  (  _I  |`  A ) )
15 fveq1 5524 . . . . . 6  |-  (.graph  =  2nd  ->  (.graph  `  ( ( Id SetCat `  U ) `  A
) )  =  ( 2nd `  ( ( Id SetCat `  U ) `  A ) ) )
1615eqeq1d 2291 . . . . 5  |-  (.graph  =  2nd  ->  ( (.graph  `  ( ( Id SetCat `  U ) `  A
) )  =  (  _I  |`  A )  <->  ( 2nd `  ( ( Id SetCat `  U ) `  A ) )  =  (  _I  |`  A ) ) )
1714, 16syl5ibr 212 . . . 4  |-  (.graph  =  2nd  ->  ( ( U  e.  Univ  /\  A  e.  U )  ->  (.graph  `  ( ( Id SetCat `  U ) `  A
) )  =  (  _I  |`  A )
) )
182, 17ax-mp 8 . . 3  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (.graph  `  ( ( Id SetCat `  U ) `  A
) )  =  (  _I  |`  A )
)
19 fveq1 5524 . . . . 5  |-  (.id  =  ( Id SetCat `  U )  ->  (.id  `  A )  =  ( ( Id SetCat `  U
) `  A )
)
2019fveq2d 5529 . . . 4  |-  (.id  =  ( Id SetCat `  U )  ->  (.graph  `  (.id  `  A
) )  =  (.graph  `  ( ( Id SetCat `  U ) `  A ) ) )
2120eqeq1d 2291 . . 3  |-  (.id  =  ( Id SetCat `  U )  ->  (
(.graph  `  (.id  `  A ) )  =  (  _I  |`  A )  <-> 
(.graph  `  (
( Id SetCat `  U
) `  A )
)  =  (  _I  |`  A ) ) )
2218, 21syl5ibr 212 . 2  |-  (.id  =  ( Id SetCat `  U )  ->  (
( U  e.  Univ  /\  A  e.  U )  ->  (.graph  `  (.id  `  A
) )  =  (  _I  |`  A )
) )
231, 22ax-mp 8 1  |-  ( ( U  e.  Univ  /\  A  e.  U )  ->  (.graph  `  (.id  `  A
) )  =  (  _I  |`  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    _I cid 4304    |` cres 4691   ` cfv 5255   2ndc2nd 6121   Univcgru 8412   Id SetCatcidcase 25939
This theorem is referenced by:  grphidmor3  25954  cmpidmor2  25969
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-rep 4131  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-2nd 6123  df-idcatset 25940
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