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Theorem yonval 15854
 Description: Value of the Yoneda embedding. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
yonval.y Yon
yonval.c
yonval.o oppCat
yonval.m HomF
Assertion
Ref Expression
yonval curryF

Proof of Theorem yonval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 yonval.y . 2 Yon
2 df-yon 15844 . . . 4 Yon oppCat curryF HomFoppCat
32a1i 11 . . 3 Yon oppCat curryF HomFoppCat
4 simpr 459 . . . . 5
54fveq2d 5853 . . . . . 6 oppCat oppCat
6 yonval.o . . . . . 6 oppCat
75, 6syl6eqr 2461 . . . . 5 oppCat
84, 7opeq12d 4167 . . . 4 oppCat
97fveq2d 5853 . . . . 5 HomFoppCat HomF
10 yonval.m . . . . 5 HomF
119, 10syl6eqr 2461 . . . 4 HomFoppCat
128, 11oveq12d 6296 . . 3 oppCat curryF HomFoppCat curryF
13 yonval.c . . 3
14 ovex 6306 . . . 4 curryF
1514a1i 11 . . 3 curryF
163, 12, 13, 15fvmptd 5938 . 2 Yon curryF
171, 16syl5eq 2455 1 curryF
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 367   wceq 1405   wcel 1842  cvv 3059  cop 3978   cmpt 4453  cfv 5569  (class class class)co 6278  ccat 15278  oppCatcoppc 15324   curryF ccurf 15803  HomFchof 15841  Yoncyon 15842 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-yon 15844 This theorem is referenced by:  yoncl  15855  yon11  15857  yon12  15858  yon2  15859  yonpropd  15861  oppcyon  15862
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