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Theorem hofval 15845
 Description: Value of the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from oppCat to , whose object part is the hom-function , and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m HomF
hofval.c
hofval.b
hofval.h
hofval.o comp
Assertion
Ref Expression
hofval f
Distinct variable groups:   ,,,,,   ,,,,,   ,,,,,   ,,,,,   ,,,,,
Allowed substitution hints:   (,,,,)

Proof of Theorem hofval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . 2 HomF
2 df-hof 15843 . . . 4 HomF f comp comp
32a1i 11 . . 3 HomF f comp comp
4 simpr 459 . . . . 5
54fveq2d 5853 . . . 4 f f
6 fvex 5859 . . . . . 6
76a1i 11 . . . . 5
84fveq2d 5853 . . . . . 6
9 hofval.b . . . . . 6
108, 9syl6eqr 2461 . . . . 5
11 simpr 459 . . . . . . 7
1211sqxpeqd 4849 . . . . . 6
13 simplr 754 . . . . . . . . . 10
1413fveq2d 5853 . . . . . . . . 9
15 hofval.h . . . . . . . . 9
1614, 15syl6eqr 2461 . . . . . . . 8
1716oveqd 6295 . . . . . . 7
1816oveqd 6295 . . . . . . 7
1916fveq1d 5851 . . . . . . . 8
2013fveq2d 5853 . . . . . . . . . . 11 comp comp
21 hofval.o . . . . . . . . . . 11 comp
2220, 21syl6eqr 2461 . . . . . . . . . 10 comp
2322oveqd 6295 . . . . . . . . 9 comp
2422oveqd 6295 . . . . . . . . . 10 comp
2524oveqd 6295 . . . . . . . . 9 comp
26 eqidd 2403 . . . . . . . . 9
2723, 25, 26oveq123d 6299 . . . . . . . 8 comp comp
2819, 27mpteq12dv 4473 . . . . . . 7 comp comp
2917, 18, 28mpt2eq123dv 6340 . . . . . 6 comp comp
3012, 12, 29mpt2eq123dv 6340 . . . . 5 comp comp
317, 10, 30csbied2 3401 . . . 4 comp comp
325, 31opeq12d 4167 . . 3 f comp comp f
33 hofval.c . . 3
34 opex 4655 . . . 4 f
3534a1i 11 . . 3 f
363, 32, 33, 35fvmptd 5938 . 2 HomF f
371, 36syl5eq 2455 1 f
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   wceq 1405   wcel 1842  cvv 3059  csb 3373  cop 3978   cmpt 4453   cxp 4821  cfv 5569  (class class class)co 6278   cmpt2 6280  c1st 6782  c2nd 6783  cbs 14841   chom 14920  compcco 14921  ccat 15278   f chomf 15280  HomFchof 15841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-hof 15843 This theorem is referenced by:  hof1fval  15846  hof2fval  15848  hofcl  15852  hofpropd  15860
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