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Theorem isinito 15396
Description: The predicate "is an initial object" of a category. (Contributed by AV, 3-Apr-2020.)
Hypotheses
Ref Expression
isinito.b  |-  B  =  ( Base `  C
)
isinito.h  |-  H  =  ( Hom  `  C
)
isinito.c  |-  ( ph  ->  C  e.  Cat )
isinito.i  |-  ( ph  ->  I  e.  B )
Assertion
Ref Expression
isinito  |-  ( ph  ->  ( I  e.  (InitO `  C )  <->  A. b  e.  B  E! h  h  e.  ( I H b ) ) )
Distinct variable groups:    B, b    C, b, h    I, b, h
Allowed substitution hints:    ph( h, b)    B( h)    H( h, b)

Proof of Theorem isinito
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 isinito.c . . . 4  |-  ( ph  ->  C  e.  Cat )
2 isinito.b . . . 4  |-  B  =  ( Base `  C
)
3 isinito.h . . . 4  |-  H  =  ( Hom  `  C
)
41, 2, 3initoval 15393 . . 3  |-  ( ph  ->  (InitO `  C )  =  { i  e.  B  |  A. b  e.  B  E! h  h  e.  ( i H b ) } )
54eleq2d 2452 . 2  |-  ( ph  ->  ( I  e.  (InitO `  C )  <->  I  e.  { i  e.  B  |  A. b  e.  B  E! h  h  e.  ( i H b ) } ) )
6 isinito.i . . 3  |-  ( ph  ->  I  e.  B )
7 oveq1 6203 . . . . . . 7  |-  ( i  =  I  ->  (
i H b )  =  ( I H b ) )
87eleq2d 2452 . . . . . 6  |-  ( i  =  I  ->  (
h  e.  ( i H b )  <->  h  e.  ( I H b ) ) )
98eubidv 2240 . . . . 5  |-  ( i  =  I  ->  ( E! h  h  e.  ( i H b )  <->  E! h  h  e.  ( I H b ) ) )
109ralbidv 2821 . . . 4  |-  ( i  =  I  ->  ( A. b  e.  B  E! h  h  e.  ( i H b )  <->  A. b  e.  B  E! h  h  e.  ( I H b ) ) )
1110elrab3 3183 . . 3  |-  ( I  e.  B  ->  (
I  e.  { i  e.  B  |  A. b  e.  B  E! h  h  e.  (
i H b ) }  <->  A. b  e.  B  E! h  h  e.  ( I H b ) ) )
126, 11syl 16 . 2  |-  ( ph  ->  ( I  e.  {
i  e.  B  |  A. b  e.  B  E! h  h  e.  ( i H b ) }  <->  A. b  e.  B  E! h  h  e.  ( I H b ) ) )
135, 12bitrd 253 1  |-  ( ph  ->  ( I  e.  (InitO `  C )  <->  A. b  e.  B  E! h  h  e.  ( I H b ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1399    e. wcel 1826   E!weu 2218   A.wral 2732   {crab 2736   ` cfv 5496  (class class class)co 6196   Basecbs 14634   Hom chom 14713   Catccat 15071  InitOcinito 15384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-inito 15387
This theorem is referenced by:  isinitoi  15399  initoeu2  15412  zrinitorngc  33008  irinitoringc  33077  zrninitoringc  33079
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