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Theorem funcf2 17814
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 7408 . . . 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 17813 . . . . 5 (𝜑𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)))
7 funcf2.x . . . . . 6 (𝜑𝑋𝐵)
8 funcf2.y . . . . . 6 (𝜑𝑌𝐵)
97, 8opelxpd 5713 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10 2fveq3 6893 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(1st𝑧)) = (𝐹‘(1st ‘⟨𝑋, 𝑌⟩)))
11 2fveq3 6893 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(2nd𝑧)) = (𝐹‘(2nd ‘⟨𝑋, 𝑌⟩)))
1210, 11oveq12d 7423 . . . . . . 7 (𝑧 = ⟨𝑋, 𝑌⟩ → ((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) = ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))))
13 fveq2 6888 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
14 df-ov 7408 . . . . . . . 8 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1513, 14eqtr4di 2790 . . . . . . 7 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻𝑧) = (𝑋𝐻𝑌))
1612, 15oveq12d 7423 . . . . . 6 (𝑧 = ⟨𝑋, 𝑌⟩ → (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) = (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
1716fvixp 8892 . . . . 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 584 . . . 4 (𝜑 → (𝐺‘⟨𝑋, 𝑌⟩) ∈ (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
191, 18eqeltrid 2837 . . 3 (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)))
20 op1stg 7983 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2120fveq2d 6892 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝐹‘(1st ‘⟨𝑋, 𝑌⟩)) = (𝐹𝑋))
22 op2ndg 7984 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2322fveq2d 6892 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝐹‘(2nd ‘⟨𝑋, 𝑌⟩)) = (𝐹𝑌))
2421, 23oveq12d 7423 . . . . 5 ((𝑋𝐵𝑌𝐵) → ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹𝑋)𝐽(𝐹𝑌)))
257, 8, 24syl2anc 584 . . . 4 (𝜑 → ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹𝑋)𝐽(𝐹𝑌)))
2625oveq1d 7420 . . 3 (𝜑 → (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑m (𝑋𝐻𝑌)) = (((𝐹𝑋)𝐽(𝐹𝑌)) ↑m (𝑋𝐻𝑌)))
2719, 26eleqtrd 2835 . 2 (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹𝑋)𝐽(𝐹𝑌)) ↑m (𝑋𝐻𝑌)))
28 elmapi 8839 . 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 396   = wceq 1541  wcel 2106  cop 4633   class class class wbr 5147   × cxp 5673  wf 6536  cfv 6540  (class class class)co 7405  1st c1st 7969  2nd c2nd 7970  m cmap 8816  Xcixp 8887  Basecbs 17140  Hom chom 17204   Func cfunc 17800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-ixp 8888  df-func 17804
This theorem is referenced by:  funcsect  17818  funcoppc  17821  cofu2  17832  cofucl  17834  cofulid  17836  cofurid  17837  funcres  17842  funcres2  17844  funcres2c  17848  isfull2  17858  isfth2  17862  fthsect  17872  fthmon  17874  fuccocl  17913  fucidcl  17914  invfuc  17923  natpropd  17925  catciso  18057  prfval  18147  prfcl  18151  prf1st  18152  prf2nd  18153  1st2ndprf  18154  evlfcllem  18170  evlfcl  18171  curf1cl  18177  curf2cl  18180  uncf2  18186  curfuncf  18187  uncfcurf  18188  diag2cl  18195  curf2ndf  18196  yonedalem4c  18226  yonedalem3b  18228  yonedainv  18230  yonffthlem  18231  fullthinc  47619  fullthinc2  47620  thincfth  47621  thincciso  47622
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