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Theorem evlfval 15360
 Description: Value of the evaluation functor. (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
evlfval.e evalF
evlfval.c
evlfval.d
evlfval.b
evlfval.h
evlfval.o comp
evlfval.n Nat
Assertion
Ref Expression
evlfval
Distinct variable groups:   ,,,,,,,   ,,,,,,,   ,,,,,   ,,,,,,   ,,,,,,,   ,,,,,,   ,,
Allowed substitution hints:   (,,,,)   ()   (,,,,,,)   (,)   ()

Proof of Theorem evlfval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlfval.e . 2 evalF
2 df-evlf 15356 . . . 4 evalF Nat comp
32a1i 11 . . 3 evalF Nat comp
4 simprl 756 . . . . . 6
5 simprr 757 . . . . . 6
64, 5oveq12d 6299 . . . . 5
74fveq2d 5860 . . . . . 6
8 evlfval.b . . . . . 6
97, 8syl6eqr 2502 . . . . 5
10 eqidd 2444 . . . . 5
116, 9, 10mpt2eq123dv 6344 . . . 4
126, 9xpeq12d 5014 . . . . 5
134, 5oveq12d 6299 . . . . . . . . . 10 Nat Nat
14 evlfval.n . . . . . . . . . 10 Nat
1513, 14syl6eqr 2502 . . . . . . . . 9 Nat
1615oveqd 6298 . . . . . . . 8 Nat
174fveq2d 5860 . . . . . . . . . 10
18 evlfval.h . . . . . . . . . 10
1917, 18syl6eqr 2502 . . . . . . . . 9
2019oveqd 6298 . . . . . . . 8
215fveq2d 5860 . . . . . . . . . . 11 comp comp
22 evlfval.o . . . . . . . . . . 11 comp
2321, 22syl6eqr 2502 . . . . . . . . . 10 comp
2423oveqd 6298 . . . . . . . . 9 comp
2524oveqd 6298 . . . . . . . 8 comp
2616, 20, 25mpt2eq123dv 6344 . . . . . . 7 Nat comp
2726csbeq2dv 3821 . . . . . 6 Nat comp
2827csbeq2dv 3821 . . . . 5 Nat comp
2912, 12, 28mpt2eq123dv 6344 . . . 4 Nat comp
3011, 29opeq12d 4210 . . 3 Nat comp
31 evlfval.c . . 3
32 evlfval.d . . 3
33 opex 4701 . . . 4
3433a1i 11 . . 3
353, 30, 31, 32, 34ovmpt2d 6415 . 2 evalF
361, 35syl5eq 2496 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1383   wcel 1804  cvv 3095  csb 3420  cop 4020   cxp 4987  cfv 5578  (class class class)co 6281   cmpt2 6283  c1st 6783  c2nd 6784  cbs 14509   chom 14585  compcco 14586  ccat 14938   cfunc 15097   Nat cnat 15184   evalF cevlf 15352 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-evlf 15356 This theorem is referenced by:  evlf2  15361  evlf1  15363  evlfcl  15365
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