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Theorem funcf2 17833

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)
funcf2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
funcf2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
funcf2	⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Proof of Theorem funcf2 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7366	. . . 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 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
6	2, 3, 4, 5	funcixp 17832	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1^st ‘𝑧))𝐽(𝐹‘(2^nd ‘𝑧))) ↑_m (𝐻‘𝑧)))
7		funcf2.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		funcf2.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	7, 8	opelxpd 5664	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10		2fveq3 6839	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(1^st ‘𝑧)) = (𝐹‘(1^st ‘⟨𝑋, 𝑌⟩)))
11		2fveq3 6839	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(2^nd ‘𝑧)) = (𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩)))
12	10, 11	oveq12d 7381	. . . . . . 7 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → ((𝐹‘(1^st ‘𝑧))𝐽(𝐹‘(2^nd ‘𝑧))) = ((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))))
13		fveq2 6834	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
14		df-ov 7366	. . . . . . . 8 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
15	13, 14	eqtr4di 2793	. . . . . . 7 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻‘𝑧) = (𝑋𝐻𝑌))
16	12, 15	oveq12d 7381	. . . . . 6 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (((𝐹‘(1^st ‘𝑧))𝐽(𝐹‘(2^nd ‘𝑧))) ↑_m (𝐻‘𝑧)) = (((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) ↑_m (𝑋𝐻𝑌)))
17	16	fvixp 8847	. . . . 5 ⊢ ((𝐺 ∈ X𝑧 ∈ (𝐵 × 𝐵)(((𝐹‘(1^st ‘𝑧))𝐽(𝐹‘(2^nd ‘𝑧))) ↑_m (𝐻‘𝑧)) ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵)) → (𝐺‘⟨𝑋, 𝑌⟩) ∈ (((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) ↑_m (𝑋𝐻𝑌)))
18	6, 9, 17	syl2anc 590	. . . 4 ⊢ (𝜑 → (𝐺‘⟨𝑋, 𝑌⟩) ∈ (((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) ↑_m (𝑋𝐻𝑌)))
19	1, 18	eqeltrid 2844	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) ↑_m (𝑋𝐻𝑌)))
20		op1stg 7950	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
21	20	fveq2d 6838	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(1^st ‘⟨𝑋, 𝑌⟩)) = (𝐹‘𝑋))
22		op2ndg 7951	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
23	22	fveq2d 6838	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩)) = (𝐹‘𝑌))
24	21, 23	oveq12d 7381	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
25	7, 8, 24	syl2anc 590	. . . 4 ⊢ (𝜑 → ((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
26	25	oveq1d 7378	. . 3 ⊢ (𝜑 → (((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) ↑_m (𝑋𝐻𝑌)) = (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↑_m (𝑋𝐻𝑌)))
27	19, 26	eleqtrd 2842	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↑_m (𝑋𝐻𝑌)))
28		elmapi 8793	. 2 ⊢ ((𝑋𝐺𝑌) ∈ (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↑_m (𝑋𝐻𝑌)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
29	27, 28	syl 17	1 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ⟨cop 4568 class class class wbr 5079 × cxp 5623 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 1^st c1st 7936 2^nd c2nd 7937 ↑_m cmap 8770 Xcixp 8842 Basecbs 17177 Hom chom 17229 Func cfunc 17819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-1st 7938 df-2nd 7939 df-map 8772 df-ixp 8843 df-func 17823

This theorem is referenced by: funcsect 17837 funcoppc 17840 cofu2 17851 cofucl 17853 cofulid 17855 cofurid 17856 funcres 17861 funcres2 17863 funcres2c 17868 isfull2 17878 isfth2 17882 fthsect 17892 fthmon 17894 fuccocl 17932 fucidcl 17933 invfuc 17942 natpropd 17944 catciso 18076 prfval 18163 prfcl 18167 prf1st 18168 prf2nd 18169 1st2ndprf 18170 evlfcllem 18185 evlfcl 18186 curf1cl 18192 curf2cl 18195 uncf2 18201 curfuncf 18202 uncfcurf 18203 diag2cl 18210 curf2ndf 18211 yonedalem4c 18241 yonedalem3b 18243 yonedainv 18245 yonffthlem 18246 funchomf 49594 cofidf2a 49614 imassc 49650 imaid 49651 imaf1co 49652 upciclem2 49664 upeu2 49669 uppropd 49678 uptrlem1 49707 uptrlem3 49709 diag1 49801 diag2f1 49806 fuco112xa 49830 fuco22natlem1 49839 fuco22natlem2 49840 fuco22natlem3 49841 fuco22natlem 49842 fucocolem1 49850 fucocolem3 49852 fucoco 49854 fucolid 49858 prcofdiag1 49890 prcofdiag 49891 oppfdiag1 49911 oppfdiag 49913 functhincfun 49946 fullthinc 49947 fullthinc2 49948 thincfth 49949 thincciso 49950 termcfuncval 50029

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