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Theorem funcf2 17772

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 7349	. . . 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 17771	. . . . 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 5655	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10		2fveq3 6827	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(1^st ‘𝑧)) = (𝐹‘(1^st ‘⟨𝑋, 𝑌⟩)))
11		2fveq3 6827	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(2^nd ‘𝑧)) = (𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩)))
12	10, 11	oveq12d 7364	. . . . . . 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 6822	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
14		df-ov 7349	. . . . . . . 8 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
15	13, 14	eqtr4di 2784	. . . . . . 7 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻‘𝑧) = (𝑋𝐻𝑌))
16	12, 15	oveq12d 7364	. . . . . 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 8826	. . . . 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 584	. . . 4 ⊢ (𝜑 → (𝐺‘⟨𝑋, 𝑌⟩) ∈ (((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) ↑_m (𝑋𝐻𝑌)))
19	1, 18	eqeltrid 2835	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) ↑_m (𝑋𝐻𝑌)))
20		op1stg 7933	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
21	20	fveq2d 6826	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(1^st ‘⟨𝑋, 𝑌⟩)) = (𝐹‘𝑋))
22		op2ndg 7934	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
23	22	fveq2d 6826	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩)) = (𝐹‘𝑌))
24	21, 23	oveq12d 7364	. . . . 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 584	. . . 4 ⊢ (𝜑 → ((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
26	25	oveq1d 7361	. . 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 2833	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↑_m (𝑋𝐻𝑌)))
28		elmapi 8773	. 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 395 = wceq 1541 ∈ wcel 2111 ⟨cop 4582 class class class wbr 5091 × cxp 5614 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 ↑_m cmap 8750 Xcixp 8821 Basecbs 17117 Hom chom 17169 Func cfunc 17758

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-1st 7921 df-2nd 7922 df-map 8752 df-ixp 8822 df-func 17762

This theorem is referenced by: funcsect 17776 funcoppc 17779 cofu2 17790 cofucl 17792 cofulid 17794 cofurid 17795 funcres 17800 funcres2 17802 funcres2c 17807 isfull2 17817 isfth2 17821 fthsect 17831 fthmon 17833 fuccocl 17871 fucidcl 17872 invfuc 17881 natpropd 17883 catciso 18015 prfval 18102 prfcl 18106 prf1st 18107 prf2nd 18108 1st2ndprf 18109 evlfcllem 18124 evlfcl 18125 curf1cl 18131 curf2cl 18134 uncf2 18140 curfuncf 18141 uncfcurf 18142 diag2cl 18149 curf2ndf 18150 yonedalem4c 18180 yonedalem3b 18182 yonedainv 18184 yonffthlem 18185 funchomf 49128 cofidf2a 49148 imassc 49184 imaid 49185 imaf1co 49186 upciclem2 49198 upeu2 49203 uppropd 49212 uptrlem1 49241 uptrlem3 49243 diag1 49335 diag2f1 49340 fuco112xa 49364 fuco22natlem1 49373 fuco22natlem2 49374 fuco22natlem3 49375 fuco22natlem 49376 fucocolem1 49384 fucocolem3 49386 fucoco 49388 fucolid 49392 prcofdiag1 49424 prcofdiag 49425 oppfdiag1 49445 oppfdiag 49447 functhincfun 49480 fullthinc 49481 fullthinc2 49482 thincfth 49483 thincciso 49484 termcfuncval 49563

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