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Theorem catcfuccl 14256
 Description: The category of categories for a weak universe is closed under the functor category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
catcfuccl.c CatCat
catcfuccl.b
catcfuccl.o FuncCat
catcfuccl.u WUni
catcfuccl.1
catcfuccl.x
catcfuccl.y
Assertion
Ref Expression
catcfuccl

Proof of Theorem catcfuccl
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catcfuccl.o . . . . 5 FuncCat
2 eqid 2435 . . . . 5
3 eqid 2435 . . . . 5 Nat Nat
4 eqid 2435 . . . . 5
5 eqid 2435 . . . . 5 comp comp
6 inss2 3554 . . . . . 6
7 catcfuccl.x . . . . . . 7
8 catcfuccl.c . . . . . . . 8 CatCat
9 catcfuccl.b . . . . . . . 8
10 catcfuccl.u . . . . . . . 8 WUni
118, 9, 10catcbas 14244 . . . . . . 7
127, 11eleqtrd 2511 . . . . . 6
136, 12sseldi 3338 . . . . 5
14 catcfuccl.y . . . . . . 7
1514, 11eleqtrd 2511 . . . . . 6
166, 15sseldi 3338 . . . . 5
17 eqidd 2436 . . . . 5 Nat Nat comp Nat Nat comp
181, 2, 3, 4, 5, 13, 16, 17fucval 14147 . . . 4 Nat comp Nat Nat comp
19 df-base 13466 . . . . . . 7 Slot
20 catcfuccl.1 . . . . . . . 8
2110, 20wunndx 13477 . . . . . . 7
2219, 10, 21wunstr 13480 . . . . . 6
23 inss1 3553 . . . . . . . 8
2423, 12sseldi 3338 . . . . . . 7
2523, 15sseldi 3338 . . . . . . 7
2610, 24, 25wunfunc 14088 . . . . . 6
2710, 22, 26wunop 8589 . . . . 5
28 df-hom 13545 . . . . . . 7 Slot ;
2928, 10, 21wunstr 13480 . . . . . 6
3010, 24, 25wunnat 14145 . . . . . 6 Nat
3110, 29, 30wunop 8589 . . . . 5 Nat
32 df-cco 13546 . . . . . . 7 comp Slot ;
3332, 10, 21wunstr 13480 . . . . . 6 comp
3410, 26, 26wunxp 8591 . . . . . . . 8
3510, 34, 26wunxp 8591 . . . . . . 7
3632, 10, 25wunstr 13480 . . . . . . . . . . . . . 14 comp
3710, 36wunrn 8596 . . . . . . . . . . . . 13 comp
3810, 37wununi 8573 . . . . . . . . . . . 12 comp
3910, 38wunrn 8596 . . . . . . . . . . 11 comp
4010, 39wununi 8573 . . . . . . . . . 10 comp
4110, 40wunpw 8574 . . . . . . . . 9 comp
4219, 10, 24wunstr 13480 . . . . . . . . 9
4310, 41, 42wunmap 8593 . . . . . . . 8 comp
4410, 30wunrn 8596 . . . . . . . . . 10 Nat
4510, 44wununi 8573 . . . . . . . . 9 Nat
4610, 45, 45wunxp 8591 . . . . . . . 8 Nat Nat
4710, 43, 46wunpm 8592 . . . . . . 7 comp Nat Nat
48 fvex 5734 . . . . . . . . . . 11
49 fvex 5734 . . . . . . . . . . . . . 14
50 ovex 6098 . . . . . . . . . . . . . . . . 17 comp
51 ovex 6098 . . . . . . . . . . . . . . . . . . . 20 Nat
5251rnex 5125 . . . . . . . . . . . . . . . . . . 19 Nat
5352uniex 4697 . . . . . . . . . . . . . . . . . 18 Nat
5453, 53xpex 4982 . . . . . . . . . . . . . . . . 17 Nat Nat
55 eqid 2435 . . . . . . . . . . . . . . . . . . . . 21 comp comp
56 ovssunirn 6099 . . . . . . . . . . . . . . . . . . . . . . . 24 comp comp
57 ovssunirn 6099 . . . . . . . . . . . . . . . . . . . . . . . . 25 comp comp
58 rnss 5090 . . . . . . . . . . . . . . . . . . . . . . . . 25 comp comp comp comp
59 uniss 4028 . . . . . . . . . . . . . . . . . . . . . . . . 25 comp comp comp comp
6057, 58, 59mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . 24 comp comp
6156, 60sstri 3349 . . . . . . . . . . . . . . . . . . . . . . 23 comp comp
62 ovex 6098 . . . . . . . . . . . . . . . . . . . . . . . 24 comp
6362elpw 3797 . . . . . . . . . . . . . . . . . . . . . . 23 comp comp comp comp
6461, 63mpbir 201 . . . . . . . . . . . . . . . . . . . . . 22 comp comp
6564a1i 11 . . . . . . . . . . . . . . . . . . . . 21 comp comp
6655, 65fmpti 5884 . . . . . . . . . . . . . . . . . . . 20 comp comp
67 fvex 5734 . . . . . . . . . . . . . . . . . . . . . . . . . 26 comp
6867rnex 5125 . . . . . . . . . . . . . . . . . . . . . . . . 25 comp
6968uniex 4697 . . . . . . . . . . . . . . . . . . . . . . . 24 comp
7069rnex 5125 . . . . . . . . . . . . . . . . . . . . . . 23 comp
7170uniex 4697 . . . . . . . . . . . . . . . . . . . . . 22 comp
7271pwex 4374 . . . . . . . . . . . . . . . . . . . . 21 comp
73 fvex 5734 . . . . . . . . . . . . . . . . . . . . 21
7472, 73elmap 7034 . . . . . . . . . . . . . . . . . . . 20 comp comp comp comp
7566, 74mpbir 201 . . . . . . . . . . . . . . . . . . 19 comp comp
7675rgen2w 2766 . . . . . . . . . . . . . . . . . 18 Nat Nat comp comp
77 eqid 2435 . . . . . . . . . . . . . . . . . . 19 Nat Nat comp Nat Nat comp
7877fmpt2 6410 . . . . . . . . . . . . . . . . . 18 Nat Nat comp comp Nat Nat comp Nat Nat comp
7976, 78mpbi 200 . . . . . . . . . . . . . . . . 17 Nat Nat comp Nat Nat comp
80 ovssunirn 6099 . . . . . . . . . . . . . . . . . 18 Nat Nat
81 ovssunirn 6099 . . . . . . . . . . . . . . . . . 18 Nat Nat
82 xpss12 4973 . . . . . . . . . . . . . . . . . 18 Nat Nat Nat Nat Nat Nat Nat Nat
8380, 81, 82mp2an 654 . . . . . . . . . . . . . . . . 17 Nat Nat Nat Nat
84 elpm2r 7026 . . . . . . . . . . . . . . . . 17 comp Nat Nat Nat Nat comp Nat Nat comp Nat Nat Nat Nat Nat Nat comp comp Nat Nat
8550, 54, 79, 83, 84mp4an 655 . . . . . . . . . . . . . . . 16 Nat Nat comp comp Nat Nat
8685sbcth 3167 . . . . . . . . . . . . . . 15 Nat Nat comp comp Nat Nat
87 sbcel1g 3262 . . . . . . . . . . . . . . 15 Nat Nat comp comp Nat Nat Nat Nat comp comp Nat Nat
8886, 87mpbid 202 . . . . . . . . . . . . . 14 Nat Nat comp comp Nat Nat
8949, 88ax-mp 8 . . . . . . . . . . . . 13 Nat Nat comp comp Nat Nat
9089sbcth 3167 . . . . . . . . . . . 12 Nat Nat comp comp Nat Nat
91 sbcel1g 3262 . . . . . . . . . . . 12 Nat Nat comp comp Nat Nat Nat Nat comp comp Nat Nat
9290, 91mpbid 202 . . . . . . . . . . 11 Nat Nat comp comp Nat Nat
9348, 92ax-mp 8 . . . . . . . . . 10 Nat Nat comp comp Nat Nat
9493rgen2w 2766 . . . . . . . . 9 Nat Nat comp comp Nat Nat
95 eqid 2435 . . . . . . . . . 10 Nat Nat comp Nat Nat comp
9695fmpt2 6410 . . . . . . . . 9 Nat Nat comp comp Nat Nat Nat Nat comp comp Nat Nat
9794, 96mpbi 200 . . . . . . . 8 Nat Nat comp comp Nat Nat
9897a1i 11 . . . . . . 7 Nat Nat comp comp Nat Nat
9910, 35, 47, 98wunf 8594 . . . . . 6 Nat Nat comp
10010, 33, 99wunop 8589 . . . . 5 comp Nat Nat comp
10110, 27, 31, 100wuntp 8578 . . . 4 Nat comp Nat Nat comp
10218, 101eqeltrd 2509 . . 3
1031, 13, 16fuccat 14159 . . 3
104 elin 3522 . . 3
105102, 103, 104sylanbrc 646 . 2
106105, 11eleqtrrd 2512 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  wral 2697  cvv 2948  wsbc 3153  csb 3243   cin 3311   wss 3312  cpw 3791  ctp 3808  cop 3809  cuni 4007   cmpt 4258  com 4837   cxp 4868   crn 4871  wf 5442  cfv 5446  (class class class)co 6073   cmpt2 6075  c1st 6339  c2nd 6340   cmap 7010   cpm 7011  WUnicwun 8567  c1 8983  c4 10043  c5 10044  ;cdc 10374  cnx 13458  cbs 13461   chom 13532  compcco 13533  ccat 13881   cfunc 14043   Nat cnat 14130   FuncCat cfuc 14131  CatCatccatc 14241 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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  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-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-er 6897  df-ec 6899  df-qs 6903  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-wun 8569  df-ni 8741  df-pli 8742  df-mi 8743  df-lti 8744  df-plpq 8777  df-mpq 8778  df-ltpq 8779  df-enq 8780  df-nq 8781  df-erq 8782  df-plq 8783  df-mq 8784  df-1nq 8785  df-rq 8786  df-ltnq 8787  df-np 8850  df-plp 8852  df-ltp 8854  df-enr 8926  df-nr 8927  df-c 8988  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-hom 13545  df-cco 13546  df-cat 13885  df-cid 13886  df-func 14047  df-nat 14132  df-fuc 14133  df-catc 14242
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