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Theorem hof1fval 16090
 Description: The object part of the Hom functor is the f operation, which is just a functionalized version of . That is, it is a two argument function, which maps to the set of morphisms from to . (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m HomF
hofval.c
Assertion
Ref Expression
hof1fval f

Proof of Theorem hof1fval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . . 3 HomF
2 hofval.c . . 3
3 eqid 2420 . . 3
4 eqid 2420 . . 3
5 eqid 2420 . . 3 comp comp
61, 2, 3, 4, 5hofval 16089 . 2 f comp comp
7 fvex 5882 . . 3 f
8 fvex 5882 . . . . 5
98, 8xpex 6600 . . . 4
109, 9mpt2ex 6875 . . 3 comp comp
117, 10op1std 6808 . 2 f comp comp f
126, 11syl 17 1 f
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1437   wcel 1867  cop 3999   cmpt 4475   cxp 4843  cfv 5592  (class class class)co 6296   cmpt2 6298  c1st 6796  c2nd 6797  cbs 15081   chom 15161  compcco 15162  ccat 15522   f chomf 15524  HomFchof 16085 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6798  df-2nd 6799  df-hof 16087 This theorem is referenced by:  hof1  16091
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