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Theorem funcf2 16575
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 6693 . . . 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 16574 . . . . 5 (𝜑𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)))
7 funcf2.x . . . . . 6 (𝜑𝑋𝐵)
8 funcf2.y . . . . . 6 (𝜑𝑌𝐵)
9 opelxpi 5182 . . . . . 6 ((𝑋𝐵𝑌𝐵) → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
107, 8, 9syl2anc 694 . . . . 5 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
11 fveq2 6229 . . . . . . . . 9 (𝑧 = ⟨𝑋, 𝑌⟩ → (1st𝑧) = (1st ‘⟨𝑋, 𝑌⟩))
1211fveq2d 6233 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(1st𝑧)) = (𝐹‘(1st ‘⟨𝑋, 𝑌⟩)))
13 fveq2 6229 . . . . . . . . 9 (𝑧 = ⟨𝑋, 𝑌⟩ → (2nd𝑧) = (2nd ‘⟨𝑋, 𝑌⟩))
1413fveq2d 6233 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(2nd𝑧)) = (𝐹‘(2nd ‘⟨𝑋, 𝑌⟩)))
1512, 14oveq12d 6708 . . . . . . 7 (𝑧 = ⟨𝑋, 𝑌⟩ → ((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) = ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))))
16 fveq2 6229 . . . . . . . 8 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
17 df-ov 6693 . . . . . . . 8 (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
1816, 17syl6eqr 2703 . . . . . . 7 (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻𝑧) = (𝑋𝐻𝑌))
1915, 18oveq12d 6708 . . . . . 6 (𝑧 = ⟨𝑋, 𝑌⟩ → (((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑𝑚 (𝐻𝑧)) = (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑𝑚 (𝑋𝐻𝑌)))
2019fvixp 7955 . . . . 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)))
216, 10, 20syl2anc 694 . . . 4 (𝜑 → (𝐺‘⟨𝑋, 𝑌⟩) ∈ (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑𝑚 (𝑋𝐻𝑌)))
221, 21syl5eqel 2734 . . 3 (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑𝑚 (𝑋𝐻𝑌)))
23 op1stg 7222 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
2423fveq2d 6233 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝐹‘(1st ‘⟨𝑋, 𝑌⟩)) = (𝐹𝑋))
25 op2ndg 7223 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
2625fveq2d 6233 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝐹‘(2nd ‘⟨𝑋, 𝑌⟩)) = (𝐹𝑌))
2724, 26oveq12d 6708 . . . . 5 ((𝑋𝐵𝑌𝐵) → ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹𝑋)𝐽(𝐹𝑌)))
287, 8, 27syl2anc 694 . . . 4 (𝜑 → ((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹𝑋)𝐽(𝐹𝑌)))
2928oveq1d 6705 . . 3 (𝜑 → (((𝐹‘(1st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2nd ‘⟨𝑋, 𝑌⟩))) ↑𝑚 (𝑋𝐻𝑌)) = (((𝐹𝑋)𝐽(𝐹𝑌)) ↑𝑚 (𝑋𝐻𝑌)))
3022, 29eleqtrd 2732 . 2 (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹𝑋)𝐽(𝐹𝑌)) ↑𝑚 (𝑋𝐻𝑌)))
31 elmapi 7921 . 2 ((𝑋𝐺𝑌) ∈ (((𝐹𝑋)𝐽(𝐹𝑌)) ↑𝑚 (𝑋𝐻𝑌)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)𝐽(𝐹𝑌)))
3230, 31syl 17 1 (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹𝑋)𝐽(𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  cop 4216   class class class wbr 4685   × cxp 5141  wf 5922  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  Xcixp 7950  Basecbs 15904  Hom chom 15999   Func cfunc 16561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-ixp 7951  df-func 16565
This theorem is referenced by:  funcsect  16579  funcoppc  16582  cofu2  16593  cofucl  16595  cofulid  16597  cofurid  16598  funcres  16603  funcres2  16605  funcres2c  16608  isfull2  16618  isfth2  16622  fthsect  16632  fthmon  16634  fuccocl  16671  fucidcl  16672  invfuc  16681  natpropd  16683  catciso  16804  prfval  16886  prfcl  16890  prf1st  16891  prf2nd  16892  1st2ndprf  16893  evlfcllem  16908  evlfcl  16909  curf1cl  16915  curf2cl  16918  uncf2  16924  curfuncf  16925  uncfcurf  16926  diag2cl  16933  curf2ndf  16934  yonedalem4c  16964  yonedalem3b  16966  yonedainv  16968  yonffthlem  16969
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