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Theorem fucpropd 15393
 Description: If two categories have the same set of objects, morphisms, and compositions, then they have the same functor categories. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
fucpropd.1 f f
fucpropd.2 compf compf
fucpropd.3 f f
fucpropd.4 compf compf
fucpropd.a
fucpropd.b
fucpropd.c
fucpropd.d
Assertion
Ref Expression
fucpropd FuncCat FuncCat

Proof of Theorem fucpropd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucpropd.1 . . . . 5 f f
2 fucpropd.2 . . . . 5 compf compf
3 fucpropd.3 . . . . 5 f f
4 fucpropd.4 . . . . 5 compf compf
5 fucpropd.a . . . . 5
6 fucpropd.b . . . . 5
7 fucpropd.c . . . . 5
8 fucpropd.d . . . . 5
91, 2, 3, 4, 5, 6, 7, 8funcpropd 15316 . . . 4
109opeq2d 4226 . . 3
111, 2, 3, 4, 5, 6, 7, 8natpropd 15392 . . . 4 Nat Nat
1211opeq2d 4226 . . 3 Nat Nat
139sqxpeqd 5034 . . . . 5
149adantr 465 . . . . 5
15 nfv 1708 . . . . . 6
16 nfcsb1v 3446 . . . . . . 7 Nat Nat comp
1716a1i 11 . . . . . 6 Nat Nat comp
18 fvex 5882 . . . . . . 7
1918a1i 11 . . . . . 6
20 nfv 1708 . . . . . . . 8
21 nfcsb1v 3446 . . . . . . . . 9 Nat Nat comp
2221a1i 11 . . . . . . . 8 Nat Nat comp
23 fvex 5882 . . . . . . . . 9
2423a1i 11 . . . . . . . 8
2511ad3antrrr 729 . . . . . . . . . . 11 Nat Nat
2625oveqd 6313 . . . . . . . . . 10 Nat Nat
2725oveqdr 6320 . . . . . . . . . 10 Nat Nat Nat
281homfeqbas 15112 . . . . . . . . . . . 12
2928ad4antr 731 . . . . . . . . . . 11 Nat Nat
30 eqid 2457 . . . . . . . . . . . 12
31 eqid 2457 . . . . . . . . . . . 12
32 eqid 2457 . . . . . . . . . . . 12 comp comp
33 eqid 2457 . . . . . . . . . . . 12 comp comp
343ad5antr 733 . . . . . . . . . . . 12 Nat Nat f f
354ad5antr 733 . . . . . . . . . . . 12 Nat Nat compf compf
36 eqid 2457 . . . . . . . . . . . . . 14
37 relfunc 15278 . . . . . . . . . . . . . . 15
38 simpllr 760 . . . . . . . . . . . . . . . 16 Nat Nat
39 simp-4r 768 . . . . . . . . . . . . . . . . . 18 Nat Nat
4039simpld 459 . . . . . . . . . . . . . . . . 17 Nat Nat
41 xp1st 6829 . . . . . . . . . . . . . . . . 17
4240, 41syl 16 . . . . . . . . . . . . . . . 16 Nat Nat
4338, 42eqeltrd 2545 . . . . . . . . . . . . . . 15 Nat Nat
44 1st2ndbr 6848 . . . . . . . . . . . . . . 15
4537, 43, 44sylancr 663 . . . . . . . . . . . . . 14 Nat Nat
4636, 30, 45funcf1 15282 . . . . . . . . . . . . 13 Nat Nat
4746ffvelrnda 6032 . . . . . . . . . . . 12 Nat Nat
48 simplr 755 . . . . . . . . . . . . . . . 16 Nat Nat
49 xp2nd 6830 . . . . . . . . . . . . . . . . 17
5040, 49syl 16 . . . . . . . . . . . . . . . 16 Nat Nat
5148, 50eqeltrd 2545 . . . . . . . . . . . . . . 15 Nat Nat
52 1st2ndbr 6848 . . . . . . . . . . . . . . 15
5337, 51, 52sylancr 663 . . . . . . . . . . . . . 14 Nat Nat
5436, 30, 53funcf1 15282 . . . . . . . . . . . . 13 Nat Nat
5554ffvelrnda 6032 . . . . . . . . . . . 12 Nat Nat
5639simprd 463 . . . . . . . . . . . . . . 15 Nat Nat
57 1st2ndbr 6848 . . . . . . . . . . . . . . 15
5837, 56, 57sylancr 663 . . . . . . . . . . . . . 14 Nat Nat
5936, 30, 58funcf1 15282 . . . . . . . . . . . . 13 Nat Nat
6059ffvelrnda 6032 . . . . . . . . . . . 12 Nat Nat
61 eqid 2457 . . . . . . . . . . . . 13 Nat Nat
62 simplrr 762 . . . . . . . . . . . . . 14 Nat Nat Nat
6361, 62nat1st2nd 15367 . . . . . . . . . . . . 13 Nat Nat Nat
64 simpr 461 . . . . . . . . . . . . 13 Nat Nat
6561, 63, 36, 31, 64natcl 15369 . . . . . . . . . . . 12 Nat Nat
66 simplrl 761 . . . . . . . . . . . . . 14 Nat Nat Nat
6761, 66nat1st2nd 15367 . . . . . . . . . . . . 13 Nat Nat Nat
6861, 67, 36, 31, 64natcl 15369 . . . . . . . . . . . 12 Nat Nat
6930, 31, 32, 33, 34, 35, 47, 55, 60, 65, 68comfeqval 15124 . . . . . . . . . . 11 Nat Nat comp comp
7029, 69mpteq12dva 4534 . . . . . . . . . 10 Nat Nat comp comp
7126, 27, 70mpt2eq123dva 6357 . . . . . . . . 9 Nat Nat comp Nat Nat comp
72 csbeq1a 3439 . . . . . . . . . 10 Nat Nat comp Nat Nat comp
7372adantl 466 . . . . . . . . 9 Nat Nat comp Nat Nat comp
7471, 73eqtrd 2498 . . . . . . . 8 Nat Nat comp Nat Nat comp
7520, 22, 24, 74csbiedf 3451 . . . . . . 7 Nat Nat comp Nat Nat comp
76 csbeq1a 3439 . . . . . . . 8 Nat Nat comp Nat Nat comp
7776adantl 466 . . . . . . 7 Nat Nat comp Nat Nat comp
7875, 77eqtrd 2498 . . . . . 6 Nat Nat comp Nat Nat comp
7915, 17, 19, 78csbiedf 3451 . . . . 5 Nat Nat comp Nat Nat comp
8013, 14, 79mpt2eq123dva 6357 . . . 4 Nat Nat comp Nat Nat comp
8180opeq2d 4226 . . 3 comp Nat Nat comp comp Nat Nat comp
8210, 12, 81tpeq123d 4126 . 2 Nat comp Nat Nat comp Nat comp Nat Nat comp
83 eqid 2457 . . 3 FuncCat FuncCat
84 eqid 2457 . . 3
85 eqidd 2458 . . 3 Nat Nat comp Nat Nat comp
8683, 84, 61, 36, 32, 5, 7, 85fucval 15374 . 2 FuncCat Nat comp Nat Nat comp
87 eqid 2457 . . 3 FuncCat FuncCat
88 eqid 2457 . . 3
89 eqid 2457 . . 3 Nat Nat
90 eqid 2457 . . 3
91 eqidd 2458 . . 3 Nat Nat comp Nat Nat comp
9287, 88, 89, 90, 33, 6, 8, 91fucval 15374 . 2 FuncCat Nat comp Nat Nat comp
9382, 86, 923eqtr4d 2508 1 FuncCat FuncCat
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395   wcel 1819  wnfc 2605  cvv 3109  csb 3430  ctp 4036  cop 4038   class class class wbr 4456   cmpt 4515   cxp 5006   wrel 5013  cfv 5594  (class class class)co 6296   cmpt2 6298  c1st 6797  c2nd 6798  cnx 14641  cbs 14644   chom 14723  compcco 14724  ccat 15081   f chomf 15083  compfccomf 15084   cfunc 15270   Nat cnat 15357   FuncCat cfuc 15358 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-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-map 7440  df-ixp 7489  df-cat 15085  df-cid 15086  df-homf 15087  df-comf 15088  df-func 15274  df-nat 15359  df-fuc 15360 This theorem is referenced by:  oyoncl  15666
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