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Theorem hof2 15380
 Description: The morphism part of the Hom functor, for morphisms (which since the first argument is contravariant means morphisms and ), yields a function (a morphism of ) mapping to . (Contributed by Mario Carneiro, 15-Jan-2017.)
Hypotheses
Ref Expression
hofval.m HomF
hofval.c
hof1.b
hof1.h
hof1.x
hof1.y
hof2.z
hof2.w
hof2.o comp
hof2.f
hof2.g
hof2.k
Assertion
Ref Expression
hof2

Proof of Theorem hof2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 hofval.m . . 3 HomF
2 hofval.c . . 3
3 hof1.b . . 3
4 hof1.h . . 3
5 hof1.x . . 3
6 hof1.y . . 3
7 hof2.z . . 3
8 hof2.w . . 3
9 hof2.o . . 3 comp
10 hof2.f . . 3
11 hof2.g . . 3
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11hof2val 15379 . 2
13 simpr 461 . . . 4
1413oveq2d 6298 . . 3
1514oveq1d 6297 . 2
16 hof2.k . 2
17 ovex 6307 . . 3
1817a1i 11 . 2
1912, 15, 16, 18fvmptd 5953 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379   wcel 1767  cvv 3113  cop 4033  cfv 5586  (class class class)co 6282  c2nd 6780  cbs 14486   chom 14562  compcco 14563  ccat 14915  HomFchof 15371 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-hof 15373 This theorem is referenced by:  yon12  15388  yon2  15389
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