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Theorem funcf2 17887
Description: The morphism part of a functor is a function on homsets. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
funcixp.b 𝐵 = (Base‘𝐷)
funcixp.h 𝐻 = (Hom ‘𝐷)
funcixp.j 𝐽 = (Hom ‘𝐸)
funcixp.f (𝜑𝐹(𝐷 Func 𝐸)𝐺)
funcf2.x (𝜑𝑋𝐵)
funcf2.y (𝜑𝑌𝐵)
Assertion
Ref Expression
funcf2 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)𝐽(𝐹𝑌)))

Proof of Theorem funcf2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ov 7427 . . . 4 (𝑋𝐺𝑌) = (𝐺‘⟨𝑋, 𝑌⟩)
2 funcixp.b . . . . . 6 𝐵 = (Base‘𝐷)
3 funcixp.h . . . . . 6 𝐻 = (Hom ‘𝐷)
4 funcixp.j . . . . . 6 𝐽 = (Hom ‘𝐸)
5 funcixp.f . . . . . 6 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
62, 3, 4, 5funcixp 17886 . . . . 5 (𝜑𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)))
7 funcf2.x . . . . . 6 (𝜑𝑋𝐵)
8 funcf2.y . . . . . 6 (𝜑𝑌𝐵)
97, 8opelxpd 5721 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10 2fveq3 6906 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(1st𝑧)) = (𝐹‘(1st ‘⟨𝑋, 𝑌⟩)))
11 2fveq3 6906 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(2nd𝑧)) = (𝐹‘(2nd ‘⟨𝑋, 𝑌⟩)))
1210, 11oveq12d 7442 . . . . . . 7 (𝑧 = ⟨𝑋, 𝑌⟩ → ((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) = ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))))
13 fveq2 6901 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
14 df-ov 7427 . . . . . . . 8 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1513, 14eqtr4di 2784 . . . . . . 7 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻𝑧) = (𝑋𝐻𝑌))
1612, 15oveq12d 7442 . . . . . 6 (𝑧 = ⟨𝑋, 𝑌⟩ → (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) = (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
1716fvixp 8931 . . . . 5 ((𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵)) → (𝐺‘⟨𝑋, 𝑌⟩) ∈ (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
186, 9, 17syl2anc 582 . . . 4 (𝜑 → (𝐺‘⟨𝑋, 𝑌⟩) ∈ (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
191, 18eqeltrid 2830 . . 3 (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
20 op1stg 8015 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2120fveq2d 6905 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝐹‘(1st ‘⟨𝑋, 𝑌⟩)) = (𝐹𝑋))
22 op2ndg 8016 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2322fveq2d 6905 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝐹‘(2nd ‘⟨𝑋, 𝑌⟩)) = (𝐹𝑌))
2421, 23oveq12d 7442 . . . . 5 ((𝑋𝐵𝑌𝐵) → ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹𝑋)𝐽(𝐹𝑌)))
257, 8, 24syl2anc 582 . . . 4 (𝜑 → ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹𝑋)𝐽(𝐹𝑌)))
2625oveq1d 7439 . . 3 (𝜑 → (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)) = (((𝐹𝑋)𝐽(𝐹𝑌)) ↑m (𝑋𝐻𝑌)))
2719, 26eleqtrd 2828 . 2 (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹𝑋)𝐽(𝐹𝑌)) ↑m (𝑋𝐻𝑌)))
28 elmapi 8878 . 2 ((𝑋𝐺𝑌) ∈ (((𝐹𝑋)𝐽(𝐹𝑌)) ↑m (𝑋𝐻𝑌)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)𝐽(𝐹𝑌)))
2927, 28syl 17 1 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  cop 4639   class class class wbr 5153   × cxp 5680  wf 6550  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  m cmap 8855  Xcixp 8926  Basecbs 17213  Hom chom 17277   Func cfunc 17873
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-map 8857  df-ixp 8927  df-func 17877
This theorem is referenced by:  funcsect  17891  funcoppc  17894  cofu2  17905  cofucl  17907  cofulid  17909  cofurid  17910  funcres  17915  funcres2  17917  funcres2c  17923  isfull2  17933  isfth2  17937  fthsect  17947  fthmon  17949  fuccocl  17989  fucidcl  17990  invfuc  17999  natpropd  18001  catciso  18133  prfval  18223  prfcl  18227  prf1st  18228  prf2nd  18229  1st2ndprf  18230  evlfcllem  18246  evlfcl  18247  curf1cl  18253  curf2cl  18256  uncf2  18262  curfuncf  18263  uncfcurf  18264  diag2cl  18271  curf2ndf  18272  yonedalem4c  18302  yonedalem3b  18304  yonedainv  18306  yonffthlem  18307  fullthinc  48367  fullthinc2  48368  thincfth  48369  thincciso  48370
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