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Theorem fucpropd 14174
 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 14097 . . . 4
109opeq2d 3991 . . 3
111, 2, 3, 4, 5, 6, 7, 8natpropd 14173 . . . 4 Nat Nat
1211opeq2d 3991 . . 3 Nat Nat
139, 9xpeq12d 4903 . . . . 5
149adantr 452 . . . . 5
15 nfv 1629 . . . . . 6
16 nfcsb1v 3283 . . . . . . 7 Nat Nat comp
1716a1i 11 . . . . . 6 Nat Nat comp
18 fvex 5742 . . . . . . 7
1918a1i 11 . . . . . 6
20 nfv 1629 . . . . . . . 8
21 nfcsb1v 3283 . . . . . . . . 9 Nat Nat comp
2221a1i 11 . . . . . . . 8 Nat Nat comp
23 fvex 5742 . . . . . . . . 9
2423a1i 11 . . . . . . . 8
2511ad3antrrr 711 . . . . . . . . . . 11 Nat Nat
2625oveqd 6098 . . . . . . . . . 10 Nat Nat
2725proplem3 13916 . . . . . . . . . 10 Nat Nat Nat
281homfeqbas 13922 . . . . . . . . . . . 12
2928ad4antr 713 . . . . . . . . . . 11 Nat Nat
30 eqid 2436 . . . . . . . . . . . 12
31 eqid 2436 . . . . . . . . . . . 12
32 eqid 2436 . . . . . . . . . . . 12 comp comp
33 eqid 2436 . . . . . . . . . . . 12 comp comp
343ad5antr 715 . . . . . . . . . . . 12 Nat Nat f f
354ad5antr 715 . . . . . . . . . . . 12 Nat Nat compf compf
36 eqid 2436 . . . . . . . . . . . . . 14
37 relfunc 14059 . . . . . . . . . . . . . . 15
38 simpllr 736 . . . . . . . . . . . . . . . 16 Nat Nat
39 simp-4r 744 . . . . . . . . . . . . . . . . . 18 Nat Nat
4039simpld 446 . . . . . . . . . . . . . . . . 17 Nat Nat
41 xp1st 6376 . . . . . . . . . . . . . . . . 17
4240, 41syl 16 . . . . . . . . . . . . . . . 16 Nat Nat
4338, 42eqeltrd 2510 . . . . . . . . . . . . . . 15 Nat Nat
44 1st2ndbr 6396 . . . . . . . . . . . . . . 15
4537, 43, 44sylancr 645 . . . . . . . . . . . . . 14 Nat Nat
4636, 30, 45funcf1 14063 . . . . . . . . . . . . 13 Nat Nat
4746ffvelrnda 5870 . . . . . . . . . . . 12 Nat Nat
48 simplr 732 . . . . . . . . . . . . . . . 16 Nat Nat
49 xp2nd 6377 . . . . . . . . . . . . . . . . 17
5040, 49syl 16 . . . . . . . . . . . . . . . 16 Nat Nat
5148, 50eqeltrd 2510 . . . . . . . . . . . . . . 15 Nat Nat
52 1st2ndbr 6396 . . . . . . . . . . . . . . 15
5337, 51, 52sylancr 645 . . . . . . . . . . . . . 14 Nat Nat
5436, 30, 53funcf1 14063 . . . . . . . . . . . . 13 Nat Nat
5554ffvelrnda 5870 . . . . . . . . . . . 12 Nat Nat
5639simprd 450 . . . . . . . . . . . . . . 15 Nat Nat
57 1st2ndbr 6396 . . . . . . . . . . . . . . 15
5837, 56, 57sylancr 645 . . . . . . . . . . . . . 14 Nat Nat
5936, 30, 58funcf1 14063 . . . . . . . . . . . . 13 Nat Nat
6059ffvelrnda 5870 . . . . . . . . . . . 12 Nat Nat
61 eqid 2436 . . . . . . . . . . . . 13 Nat Nat
62 simplrr 738 . . . . . . . . . . . . . 14 Nat Nat Nat
6361, 62nat1st2nd 14148 . . . . . . . . . . . . 13 Nat Nat Nat
64 simpr 448 . . . . . . . . . . . . 13 Nat Nat
6561, 63, 36, 31, 64natcl 14150 . . . . . . . . . . . 12 Nat Nat
66 simplrl 737 . . . . . . . . . . . . . 14 Nat Nat Nat
6761, 66nat1st2nd 14148 . . . . . . . . . . . . 13 Nat Nat Nat
6861, 67, 36, 31, 64natcl 14150 . . . . . . . . . . . 12 Nat Nat
6930, 31, 32, 33, 34, 35, 47, 55, 60, 65, 68comfeqval 13934 . . . . . . . . . . 11 Nat Nat comp comp
7029, 69mpteq12dva 4286 . . . . . . . . . 10 Nat Nat comp comp
7126, 27, 70mpt2eq123dva 6135 . . . . . . . . 9 Nat Nat comp Nat Nat comp
72 csbeq1a 3259 . . . . . . . . . 10 Nat Nat comp Nat Nat comp
7372adantl 453 . . . . . . . . 9 Nat Nat comp Nat Nat comp
7471, 73eqtrd 2468 . . . . . . . 8 Nat Nat comp Nat Nat comp
7520, 22, 24, 74csbiedf 3288 . . . . . . 7 Nat Nat comp Nat Nat comp
76 csbeq1a 3259 . . . . . . . 8 Nat Nat comp Nat Nat comp
7776adantl 453 . . . . . . 7 Nat Nat comp Nat Nat comp
7875, 77eqtrd 2468 . . . . . 6 Nat Nat comp Nat Nat comp
7915, 17, 19, 78csbiedf 3288 . . . . 5 Nat Nat comp Nat Nat comp
8013, 14, 79mpt2eq123dva 6135 . . . 4 Nat Nat comp Nat Nat comp
8180opeq2d 3991 . . 3 comp Nat Nat comp comp Nat Nat comp
8210, 12, 81tpeq123d 3898 . 2 Nat comp Nat Nat comp Nat comp Nat Nat comp
83 eqid 2436 . . 3 FuncCat FuncCat
84 eqid 2436 . . 3
85 eqidd 2437 . . 3 Nat Nat comp Nat Nat comp
8683, 84, 61, 36, 32, 5, 7, 85fucval 14155 . 2 FuncCat Nat comp Nat Nat comp
87 eqid 2436 . . 3 FuncCat FuncCat
88 eqid 2436 . . 3
89 eqid 2436 . . 3 Nat Nat
90 eqid 2436 . . 3
91 eqidd 2437 . . 3 Nat Nat comp Nat Nat comp
9287, 88, 89, 90, 33, 6, 8, 91fucval 14155 . 2 FuncCat Nat comp Nat Nat comp
9382, 86, 923eqtr4d 2478 1 FuncCat FuncCat
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wnfc 2559  cvv 2956  csb 3251  ctp 3816  cop 3817   class class class wbr 4212   cmpt 4266   cxp 4876   wrel 4883  cfv 5454  (class class class)co 6081   cmpt2 6083  c1st 6347  c2nd 6348  cnx 13466  cbs 13469   chom 13540  compcco 13541  ccat 13889   f chomf 13891  compfccomf 13892   cfunc 14051   Nat cnat 14138   FuncCat cfuc 14139 This theorem is referenced by:  oyoncl  14367 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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 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-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-map 7020  df-ixp 7064  df-cat 13893  df-cid 13894  df-homf 13895  df-comf 13896  df-func 14055  df-nat 14140  df-fuc 14141
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