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Theorem funcixp 17799
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)
Assertion
Ref Expression
funcixp (𝜑𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)))
Distinct variable groups:   𝑧,𝐵   𝑧,𝐷   𝑧,𝐸   𝜑,𝑧   𝑧,𝐹   𝑧,𝐺   𝑧,𝐽   𝑧,𝐻

Proof of Theorem funcixp
Dummy variables 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcixp.f . . 3 (𝜑𝐹(𝐷 Func 𝐸)𝐺)
2 funcixp.b . . . 4 𝐵 = (Base‘𝐷)
3 eqid 2731 . . . 4 (Base‘𝐸) = (Base‘𝐸)
4 funcixp.h . . . 4 𝐻 = (Hom ‘𝐷)
5 funcixp.j . . . 4 𝐽 = (Hom ‘𝐸)
6 eqid 2731 . . . 4 (Id‘𝐷) = (Id‘𝐷)
7 eqid 2731 . . . 4 (Id‘𝐸) = (Id‘𝐸)
8 eqid 2731 . . . 4 (comp‘𝐷) = (comp‘𝐷)
9 eqid 2731 . . . 4 (comp‘𝐸) = (comp‘𝐸)
10 df-br 5142 . . . . . . 7 (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
111, 10sylib 217 . . . . . 6 (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12 funcrcl 17795 . . . . . 6 (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1311, 12syl 17 . . . . 5 (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
1413simpld 495 . . . 4 (𝜑𝐷 ∈ Cat)
1513simprd 496 . . . 4 (𝜑𝐸 ∈ Cat)
162, 3, 4, 5, 6, 7, 8, 9, 14, 15isfunc 17796 . . 3 (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐸)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚))))))
171, 16mpbid 231 . 2 (𝜑 → (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)) ∧ ∀𝑥𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹𝑥)) ∧ ∀𝑦𝐵𝑧𝐵𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹𝑥), (𝐹𝑦)⟩(comp‘𝐸)(𝐹𝑧))((𝑥𝐺𝑦)‘𝑚)))))
1817simp2d 1143 1 (𝜑𝐺X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st𝑧))𝐽(𝐹‘(2nd𝑧))) ↑m (𝐻𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3060  cop 4628   class class class wbr 5141   × cxp 5667  wf 6528  cfv 6532  (class class class)co 7393  1st c1st 7955  2nd c2nd 7956  m cmap 8803  Xcixp 8874  Basecbs 17126  Hom chom 17190  compcco 17191  Catccat 17590  Idccid 17591   Func cfunc 17786
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-map 8805  df-ixp 8875  df-func 17790
This theorem is referenced by:  funcf2  17800  funcfn2  17801  wunfunc  17831  wunfuncOLD  17832
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