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Theorem catcval 14243
 Description: Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
catcval.c CatCat
catcval.u
catcval.b
catcval.h
catcval.o func
Assertion
Ref Expression
catcval comp
Distinct variable groups:   ,,,,   ,,,,   ,,,,   ,,,,,
Allowed substitution hints:   (,)   (,)   (,,,,,)   (,,,,,)   (,)   (,,,,,)   (,,,,,)

Proof of Theorem catcval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcval.c . 2 CatCat
2 df-catc 14242 . . . 4 CatCat comp func
32a1i 11 . . 3 CatCat comp func
4 vex 2951 . . . . . 6
54inex1 4336 . . . . 5
65a1i 11 . . . 4
7 simpr 448 . . . . . 6
87ineq1d 3533 . . . . 5
9 catcval.b . . . . . 6
109adantr 452 . . . . 5
118, 10eqtr4d 2470 . . . 4
12 simpr 448 . . . . . 6
1312opeq2d 3983 . . . . 5
14 eqidd 2436 . . . . . . . 8
1512, 12, 14mpt2eq123dv 6128 . . . . . . 7
16 catcval.h . . . . . . . 8
1716ad2antrr 707 . . . . . . 7
1815, 17eqtr4d 2470 . . . . . 6
1918opeq2d 3983 . . . . 5
2012, 12xpeq12d 4895 . . . . . . . 8
21 eqidd 2436 . . . . . . . 8 func func
2220, 12, 21mpt2eq123dv 6128 . . . . . . 7 func func
23 catcval.o . . . . . . . 8 func
2423ad2antrr 707 . . . . . . 7 func
2522, 24eqtr4d 2470 . . . . . 6 func
2625opeq2d 3983 . . . . 5 comp func comp
2713, 19, 26tpeq123d 3890 . . . 4 comp func comp
286, 11, 27csbied2 3286 . . 3 comp func comp
29 catcval.u . . . 4
30 elex 2956 . . . 4
3129, 30syl 16 . . 3
32 tpex 4700 . . . 4 comp
3332a1i 11 . . 3 comp
343, 28, 31, 33fvmptd 5802 . 2 CatCat comp
351, 34syl5eq 2479 1 comp
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cvv 2948  csb 3243   cin 3311  ctp 3808  cop 3809   cmpt 4258   cxp 4868  cfv 5446  (class class class)co 6073   cmpt2 6075  c2nd 6340  cnx 13458  cbs 13461   chom 13532  compcco 13533  ccat 13881   cfunc 14043   func ccofu 14045  CatCatccatc 14241 This theorem is referenced by:  catcbas  14244  catchomfval  14245  catccofval  14247 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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693 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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-catc 14242
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