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Theorem catidex 13899
 Description: Each object in a category has an associated identity arrow. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
catidex.b
catidex.h
catidex.o comp
catidex.c
catidex.x
Assertion
Ref Expression
catidex
Distinct variable groups:   ,,,   ,,,   ,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,)

Proof of Theorem catidex
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catidex.x . 2
2 catidex.c . . 3
3 catidex.b . . . . 5
4 catidex.h . . . . 5
5 catidex.o . . . . 5 comp
63, 4, 5iscat 13897 . . . 4
76ibi 233 . . 3
8 simpl 444 . . . 4
98ralimi 2781 . . 3
102, 7, 93syl 19 . 2
11 id 20 . . . . 5
1211, 11oveq12d 6099 . . . 4
13 oveq2 6089 . . . . . . 7
14 opeq2 3985 . . . . . . . . . 10
1514, 11oveq12d 6099 . . . . . . . . 9
1615oveqd 6098 . . . . . . . 8
1716eqeq1d 2444 . . . . . . 7
1813, 17raleqbidv 2916 . . . . . 6
19 oveq1 6088 . . . . . . 7
2011, 11opeq12d 3992 . . . . . . . . . 10
2120oveq1d 6096 . . . . . . . . 9
2221oveqd 6098 . . . . . . . 8
2322eqeq1d 2444 . . . . . . 7
2419, 23raleqbidv 2916 . . . . . 6
2518, 24anbi12d 692 . . . . 5
2625ralbidv 2725 . . . 4
2712, 26rexeqbidv 2917 . . 3
2827rspcv 3048 . 2
291, 10, 28sylc 58 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2705  wrex 2706  cop 3817  cfv 5454  (class class class)co 6081  cbs 13469   chom 13540  compcco 13541  ccat 13889 This theorem is referenced by:  catideu  13900 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084  df-cat 13893
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