Theorem funcixp 17828

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	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)

Assertion

Ref	Expression
funcixp	⊢ (𝜑 → 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)))

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.

Step	Hyp	Ref	Expression
1		funcixp.f	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
2		funcixp.b	. . . 4 ⊢ 𝐵 = (Base‘𝐷)
3		eqid 2736	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		funcixp.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
5		funcixp.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐸)
6		eqid 2736	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2736	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
8		eqid 2736	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
9		eqid 2736	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
10		df-br 5076	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11	1, 10	sylib 219	. . . . . 6 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12		funcrcl 17824	. . . . . 6 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
14	13	simpld 495	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
15	13	simprd 496	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
16	2, 3, 4, 5, 6, 7, 8, 9, 14, 15	isfunc 17825	. . 3 ⊢ (𝜑 → (𝐹(𝐷 Func 𝐸)𝐺 ↔ (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚))))))
17	1, 16	mpbid 233	. 2 ⊢ (𝜑 → (𝐹:𝐵⟶(Base‘𝐸) ∧ 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)) ∧ ∀𝑥 ∈ 𝐵 (((𝑥𝐺𝑥)‘((Id‘𝐷)‘𝑥)) = ((Id‘𝐸)‘(𝐹‘𝑥)) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑚 ∈ (𝑥𝐻𝑦)∀𝑛 ∈ (𝑦𝐻𝑧)((𝑥𝐺𝑧)‘(𝑛(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑚)) = (((𝑦𝐺𝑧)‘𝑛)(⟨(𝐹‘𝑥), (𝐹‘𝑦)⟩(comp‘𝐸)(𝐹‘𝑧))((𝑥𝐺𝑦)‘𝑚)))))
18	17	simp2d 1145	1 ⊢ (𝜑 → 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑m (𝐻‘𝑧)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1088   = wceq 1543   ∈ wcel 2115  ∀wral 3050  ⟨cop 4564   class class class wbr 5075   × cxp 5619  ⟶wf 6484  ‘cfv 6488  (class class class)co 7359  1^st c1st 7932  2^nd c2nd 7933   ↑_m cmap 8766  Xcixp 8838  Basecbs 17173  Hom chom 17225  compcco 17226  Catccat 17624  Idccid 17625   Func cfunc 17815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1970  ax-7 2011  ax-8 2117  ax-9 2125  ax-10 2148  ax-11 2164  ax-12 2185  ax-ext 2708  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7681

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 850  df-3an 1090  df-tru 1546  df-fal 1556  df-ex 1783  df-nf 1787  df-sb 2070  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2932  df-ral 3051  df-rex 3061  df-rab 3389  df-v 3430  df-sbc 3727  df-csb 3835  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-map 8768  df-ixp 8839  df-func 17819

This theorem is referenced by:  funcf2  17829  funcfn2  17830  wunfunc  17862