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Theorem catidex 15091
 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 15089 . . . 4
76ibi 241 . . 3
8 simpl 457 . . . 4
98ralimi 2850 . . 3
102, 7, 93syl 20 . 2
11 id 22 . . . . 5
1211, 11oveq12d 6314 . . . 4
13 oveq2 6304 . . . . . . 7
14 opeq2 4220 . . . . . . . . . 10
1514, 11oveq12d 6314 . . . . . . . . 9
1615oveqd 6313 . . . . . . . 8
1716eqeq1d 2459 . . . . . . 7
1813, 17raleqbidv 3068 . . . . . 6
19 oveq1 6303 . . . . . . 7
2011, 11opeq12d 4227 . . . . . . . . . 10
2120oveq1d 6311 . . . . . . . . 9
2221oveqd 6313 . . . . . . . 8
2322eqeq1d 2459 . . . . . . 7
2419, 23raleqbidv 3068 . . . . . 6
2518, 24anbi12d 710 . . . . 5
2625ralbidv 2896 . . . 4
2712, 26rexeqbidv 3069 . . 3
2827rspcv 3206 . 2
291, 10, 28sylc 60 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395   wcel 1819  wral 2807  wrex 2808  cop 4038  cfv 5594  (class class class)co 6296  cbs 14644   chom 14723  compcco 14724  ccat 15081 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-nul 4586 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-ov 6299  df-cat 15085 This theorem is referenced by:  catideu  15092
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