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Theorem funcf2 17374
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 7216 . . . 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 17373 . . . . 5 (𝜑𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)))
7 funcf2.x . . . . . 6 (𝜑𝑋𝐵)
8 funcf2.y . . . . . 6 (𝜑𝑌𝐵)
97, 8opelxpd 5589 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10 2fveq3 6722 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(1st𝑧)) = (𝐹‘(1st ‘⟨𝑋, 𝑌⟩)))
11 2fveq3 6722 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(2nd𝑧)) = (𝐹‘(2nd ‘⟨𝑋, 𝑌⟩)))
1210, 11oveq12d 7231 . . . . . . 7 (𝑧 = ⟨𝑋, 𝑌⟩ → ((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) = ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))))
13 fveq2 6717 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
14 df-ov 7216 . . . . . . . 8 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1513, 14eqtr4di 2796 . . . . . . 7 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻𝑧) = (𝑋𝐻𝑌))
1612, 15oveq12d 7231 . . . . . 6 (𝑧 = ⟨𝑋, 𝑌⟩ → (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) = (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
1716fvixp 8583 . . . . 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 587 . . . 4 (𝜑 → (𝐺‘⟨𝑋, 𝑌⟩) ∈ (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
191, 18eqeltrid 2842 . . 3 (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
20 op1stg 7773 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2120fveq2d 6721 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝐹‘(1st ‘⟨𝑋, 𝑌⟩)) = (𝐹𝑋))
22 op2ndg 7774 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2322fveq2d 6721 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝐹‘(2nd ‘⟨𝑋, 𝑌⟩)) = (𝐹𝑌))
2421, 23oveq12d 7231 . . . . 5 ((𝑋𝐵𝑌𝐵) → ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹𝑋)𝐽(𝐹𝑌)))
257, 8, 24syl2anc 587 . . . 4 (𝜑 → ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹𝑋)𝐽(𝐹𝑌)))
2625oveq1d 7228 . . 3 (𝜑 → (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)) = (((𝐹𝑋)𝐽(𝐹𝑌)) ↑m (𝑋𝐻𝑌)))
2719, 26eleqtrd 2840 . 2 (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹𝑋)𝐽(𝐹𝑌)) ↑m (𝑋𝐻𝑌)))
28 elmapi 8530 . 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 399   = wceq 1543  wcel 2110  cop 4547   class class class wbr 5053   × cxp 5549  wf 6376  cfv 6380  (class class class)co 7213  1st c1st 7759  2nd c2nd 7760  m cmap 8508  Xcixp 8578  Basecbs 16760  Hom chom 16813   Func cfunc 17360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-map 8510  df-ixp 8579  df-func 17364
This theorem is referenced by:  funcsect  17378  funcoppc  17381  cofu2  17392  cofucl  17394  cofulid  17396  cofurid  17397  funcres  17402  funcres2  17404  funcres2c  17408  isfull2  17418  isfth2  17422  fthsect  17432  fthmon  17434  fuccocl  17473  fucidcl  17474  invfuc  17483  natpropd  17485  catciso  17617  prfval  17706  prfcl  17710  prf1st  17711  prf2nd  17712  1st2ndprf  17713  evlfcllem  17729  evlfcl  17730  curf1cl  17736  curf2cl  17739  uncf2  17745  curfuncf  17746  uncfcurf  17747  diag2cl  17754  curf2ndf  17755  yonedalem4c  17785  yonedalem3b  17787  yonedainv  17789  yonffthlem  17790  fullthinc  46000  fullthinc2  46001  thincfth  46002  thincciso  46003
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