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Theorem hof2fval 15650
 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
Assertion
Ref Expression
hof2fval
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem hof2fval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . . . 4 HomF
2 hofval.c . . . 4
3 hof1.b . . . 4
4 hof1.h . . . 4
5 hof2.o . . . 4 comp
61, 2, 3, 4, 5hofval 15647 . . 3 f
7 fvex 5882 . . . 4 f
8 fvex 5882 . . . . . . 7
93, 8eqeltri 2541 . . . . . 6
109, 9xpex 6603 . . . . 5
1110, 10mpt2ex 6876 . . . 4
127, 11op2ndd 6810 . . 3 f
136, 12syl 16 . 2
14 simprr 757 . . . . . 6
1514fveq2d 5876 . . . . 5
16 hof2.z . . . . . . 7
17 hof2.w . . . . . . 7
18 op1stg 6811 . . . . . . 7
1916, 17, 18syl2anc 661 . . . . . 6
2019adantr 465 . . . . 5
2115, 20eqtrd 2498 . . . 4
22 simprl 756 . . . . . 6
2322fveq2d 5876 . . . . 5
24 hof1.x . . . . . . 7
25 hof1.y . . . . . . 7
26 op1stg 6811 . . . . . . 7
2724, 25, 26syl2anc 661 . . . . . 6
2827adantr 465 . . . . 5
2923, 28eqtrd 2498 . . . 4
3021, 29oveq12d 6314 . . 3
3122fveq2d 5876 . . . . 5
32 op2ndg 6812 . . . . . . 7
3324, 25, 32syl2anc 661 . . . . . 6
3433adantr 465 . . . . 5
3531, 34eqtrd 2498 . . . 4
3614fveq2d 5876 . . . . 5
37 op2ndg 6812 . . . . . . 7
3816, 17, 37syl2anc 661 . . . . . 6
3938adantr 465 . . . . 5
4036, 39eqtrd 2498 . . . 4
4135, 40oveq12d 6314 . . 3
4222fveq2d 5876 . . . . 5
43 df-ov 6299 . . . . 5
4442, 43syl6eqr 2516 . . . 4
4521, 29opeq12d 4227 . . . . . 6
4645, 40oveq12d 6314 . . . . 5
4722, 40oveq12d 6314 . . . . . 6
4847oveqd 6313 . . . . 5
49 eqidd 2458 . . . . 5
5046, 48, 49oveq123d 6317 . . . 4
5144, 50mpteq12dv 4535 . . 3
5230, 41, 51mpt2eq123dv 6358 . 2
53 opelxpi 5040 . . 3
5424, 25, 53syl2anc 661 . 2
55 opelxpi 5040 . . 3
5616, 17, 55syl2anc 661 . 2
57 ovex 6324 . . . 4
58 ovex 6324 . . . 4
5957, 58mpt2ex 6876 . . 3
6059a1i 11 . 2
6113, 52, 54, 56, 60ovmpt2d 6429 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395   wcel 1819  cvv 3109  cop 4038   cmpt 4515   cxp 5006  cfv 5594  (class class class)co 6296   cmpt2 6298  c1st 6797  c2nd 6798  cbs 14643   chom 14722  compcco 14723  ccat 15080   f chomf 15082  HomFchof 15643 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-hof 15645 This theorem is referenced by:  hof2val  15651
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