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Theorem fucval 14155
 Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucval.q FuncCat
fucval.b
fucval.n Nat
fucval.a
fucval.o comp
fucval.c
fucval.d
fucval.x
Assertion
Ref Expression
fucval comp
Distinct variable groups:   ,,   ,,,,,,,   ,,,,,,,   ,,,,,,,
Allowed substitution hints:   (,,,,,,)   (,,,,)   (,,,,,,)   (,,,,,,)   (,,,,,,)   (,,,,,,)

Proof of Theorem fucval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucval.q . 2 FuncCat
2 df-fuc 14141 . . . 4 FuncCat Nat comp Nat Nat comp
32a1i 11 . . 3 FuncCat Nat comp Nat Nat comp
4 simprl 733 . . . . . . 7
5 simprr 734 . . . . . . 7
64, 5oveq12d 6099 . . . . . 6
7 fucval.b . . . . . 6
86, 7syl6eqr 2486 . . . . 5
98opeq2d 3991 . . . 4
104, 5oveq12d 6099 . . . . . 6 Nat Nat
11 fucval.n . . . . . 6 Nat
1210, 11syl6eqr 2486 . . . . 5 Nat
1312opeq2d 3991 . . . 4 Nat
148, 8xpeq12d 4903 . . . . . . 7
1512oveqd 6098 . . . . . . . . . 10 Nat
1612oveqd 6098 . . . . . . . . . 10 Nat
174fveq2d 5732 . . . . . . . . . . . 12
18 fucval.a . . . . . . . . . . . 12
1917, 18syl6eqr 2486 . . . . . . . . . . 11
205fveq2d 5732 . . . . . . . . . . . . . 14 comp comp
21 fucval.o . . . . . . . . . . . . . 14 comp
2220, 21syl6eqr 2486 . . . . . . . . . . . . 13 comp
2322oveqd 6098 . . . . . . . . . . . 12 comp
2423oveqd 6098 . . . . . . . . . . 11 comp
2519, 24mpteq12dv 4287 . . . . . . . . . 10 comp
2615, 16, 25mpt2eq123dv 6136 . . . . . . . . 9 Nat Nat comp
2726csbeq2dv 3276 . . . . . . . 8 Nat Nat comp
2827csbeq2dv 3276 . . . . . . 7 Nat Nat comp
2914, 8, 28mpt2eq123dv 6136 . . . . . 6 Nat Nat comp
30 fucval.x . . . . . . 7
3130adantr 452 . . . . . 6
3229, 31eqtr4d 2471 . . . . 5 Nat Nat comp
3332opeq2d 3991 . . . 4 comp Nat Nat comp comp
349, 13, 33tpeq123d 3898 . . 3 Nat comp Nat Nat comp comp
35 fucval.c . . 3
36 fucval.d . . 3
37 tpex 4708 . . . 4 comp
3837a1i 11 . . 3 comp
393, 34, 35, 36, 38ovmpt2d 6201 . 2 FuncCat comp
401, 39syl5eq 2480 1 comp
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cvv 2956  csb 3251  ctp 3816  cop 3817   cmpt 4266   cxp 4876  cfv 5454  (class class class)co 6081   cmpt2 6083  c1st 6347  c2nd 6348  cnx 13466  cbs 13469   chom 13540  compcco 13541  ccat 13889   cfunc 14051   Nat cnat 14138   FuncCat cfuc 14139 This theorem is referenced by:  fuccofval  14156  fucbas  14157  fuchom  14158  fucpropd  14174  catcfuccl  14264 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-sep 4330  ax-nul 4338  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-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-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  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-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-fuc 14141
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