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Theorem funcf2 17854
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 7423 . . . 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 17853 . . . . 5 (𝜑𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)))
7 funcf2.x . . . . . 6 (𝜑𝑋𝐵)
8 funcf2.y . . . . . 6 (𝜑𝑌𝐵)
97, 8opelxpd 5717 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10 2fveq3 6902 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(1st𝑧)) = (𝐹‘(1st ‘⟨𝑋, 𝑌⟩)))
11 2fveq3 6902 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(2nd𝑧)) = (𝐹‘(2nd ‘⟨𝑋, 𝑌⟩)))
1210, 11oveq12d 7438 . . . . . . 7 (𝑧 = ⟨𝑋, 𝑌⟩ → ((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) = ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))))
13 fveq2 6897 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
14 df-ov 7423 . . . . . . . 8 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1513, 14eqtr4di 2786 . . . . . . 7 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻𝑧) = (𝑋𝐻𝑌))
1612, 15oveq12d 7438 . . . . . 6 (𝑧 = ⟨𝑋, 𝑌⟩ → (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) = (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
1716fvixp 8921 . . . . 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 583 . . . 4 (𝜑 → (𝐺‘⟨𝑋, 𝑌⟩) ∈ (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
191, 18eqeltrid 2833 . . 3 (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
20 op1stg 8005 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2120fveq2d 6901 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝐹‘(1st ‘⟨𝑋, 𝑌⟩)) = (𝐹𝑋))
22 op2ndg 8006 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2322fveq2d 6901 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝐹‘(2nd ‘⟨𝑋, 𝑌⟩)) = (𝐹𝑌))
2421, 23oveq12d 7438 . . . . 5 ((𝑋𝐵𝑌𝐵) → ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹𝑋)𝐽(𝐹𝑌)))
257, 8, 24syl2anc 583 . . . 4 (𝜑 → ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹𝑋)𝐽(𝐹𝑌)))
2625oveq1d 7435 . . 3 (𝜑 → (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)) = (((𝐹𝑋)𝐽(𝐹𝑌)) ↑m (𝑋𝐻𝑌)))
2719, 26eleqtrd 2831 . 2 (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹𝑋)𝐽(𝐹𝑌)) ↑m (𝑋𝐻𝑌)))
28 elmapi 8868 . 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 395   = wceq 1534  wcel 2099  cop 4635   class class class wbr 5148   × cxp 5676  wf 6544  cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  m cmap 8845  Xcixp 8916  Basecbs 17180  Hom chom 17244   Func cfunc 17840
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 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-map 8847  df-ixp 8917  df-func 17844
This theorem is referenced by:  funcsect  17858  funcoppc  17861  cofu2  17872  cofucl  17874  cofulid  17876  cofurid  17877  funcres  17882  funcres2  17884  funcres2c  17890  isfull2  17900  isfth2  17904  fthsect  17914  fthmon  17916  fuccocl  17956  fucidcl  17957  invfuc  17966  natpropd  17968  catciso  18100  prfval  18190  prfcl  18194  prf1st  18195  prf2nd  18196  1st2ndprf  18197  evlfcllem  18213  evlfcl  18214  curf1cl  18220  curf2cl  18223  uncf2  18229  curfuncf  18230  uncfcurf  18231  diag2cl  18238  curf2ndf  18239  yonedalem4c  18269  yonedalem3b  18271  yonedainv  18273  yonffthlem  18274  fullthinc  48052  fullthinc2  48053  thincfth  48054  thincciso  48055
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