Theorem funcixp 16730

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 2771	. . . 4 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		funcixp.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐷)
5		funcixp.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐸)
6		eqid 2771	. . . 4 ⊢ (Id‘𝐷) = (Id‘𝐷)
7		eqid 2771	. . . 4 ⊢ (Id‘𝐸) = (Id‘𝐸)
8		eqid 2771	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
9		eqid 2771	. . . 4 ⊢ (comp‘𝐸) = (comp‘𝐸)
10		df-br 4787	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
11	1, 10	sylib 208	. . . . . 6 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
12		funcrcl 16726	. . . . . 6 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸) → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
13	11, 12	syl 17	. . . . 5 ⊢ (𝜑 → (𝐷 ∈ Cat ∧ 𝐸 ∈ Cat))
14	13	simpld 482	. . . 4 ⊢ (𝜑 → 𝐷 ∈ Cat)
15	13	simprd 483	. . . 4 ⊢ (𝜑 → 𝐸 ∈ Cat)
16	2, 3, 4, 5, 6, 7, 8, 9, 14, 15	isfunc 16727	. . 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 222	. 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 1137	1 ⊢ (𝜑 → 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1st ‘𝑧))𝐽(𝐹‘(2nd ‘𝑧))) ↑𝑚 (𝐻‘𝑧)))

Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ⟨cop 4322   class class class wbr 4786   × cxp 5247  ⟶wf 6025  ‘cfv 6029  (class class class)co 6792  1^st c1st 7313  2^nd c2nd 7314   ↑_𝑚 cmap 8009  Xcixp 8062  Basecbs 16060  Hom chom 16156  compcco 16157  Catccat 16528  Idccid 16529   Func cfunc 16717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8011  df-ixp 8063  df-func 16721

This theorem is referenced by:  funcf2  16731  funcfn2  16732  wunfunc  16762