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Theorem funcf2 17913

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 7434	. . . 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 17912	. . . . 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 5724	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10		2fveq3 6911	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(1^st ‘𝑧)) = (𝐹‘(1^st ‘⟨𝑋, 𝑌⟩)))
11		2fveq3 6911	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(2^nd ‘𝑧)) = (𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩)))
12	10, 11	oveq12d 7449	. . . . . . 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 6906	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
14		df-ov 7434	. . . . . . . 8 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
15	13, 14	eqtr4di 2795	. . . . . . 7 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻‘𝑧) = (𝑋𝐻𝑌))
16	12, 15	oveq12d 7449	. . . . . 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 8942	. . . . 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 2845	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) ↑_m (𝑋𝐻𝑌)))
20		op1stg 8026	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
21	20	fveq2d 6910	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(1^st ‘⟨𝑋, 𝑌⟩)) = (𝐹‘𝑋))
22		op2ndg 8027	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
23	22	fveq2d 6910	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩)) = (𝐹‘𝑌))
24	21, 23	oveq12d 7449	. . . . 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 7446	. . 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 2843	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↑_m (𝑋𝐻𝑌)))
28		elmapi 8889	. 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 1540 ∈ wcel 2108 ⟨cop 4632 class class class wbr 5143 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1^st c1st 8012 2^nd c2nd 8013 ↑_m cmap 8866 Xcixp 8937 Basecbs 17247 Hom chom 17308 Func cfunc 17899

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 df-ixp 8938 df-func 17903

This theorem is referenced by: funcsect 17917 funcoppc 17920 cofu2 17931 cofucl 17933 cofulid 17935 cofurid 17936 funcres 17941 funcres2 17943 funcres2c 17948 isfull2 17958 isfth2 17962 fthsect 17972 fthmon 17974 fuccocl 18012 fucidcl 18013 invfuc 18022 natpropd 18024 catciso 18156 prfval 18244 prfcl 18248 prf1st 18249 prf2nd 18250 1st2ndprf 18251 evlfcllem 18266 evlfcl 18267 curf1cl 18273 curf2cl 18276 uncf2 18282 curfuncf 18283 uncfcurf 18284 diag2cl 18291 curf2ndf 18292 yonedalem4c 18322 yonedalem3b 18324 yonedainv 18326 yonffthlem 18327 upciclem2 48924 upeu2 48929 diag1 49004 fuco112xa 49028 fuco22natlem1 49037 fuco22natlem2 49038 fuco22natlem3 49039 fuco22natlem 49040 fucocolem1 49048 fucocolem3 49050 fucoco 49052 fucolid 49056 functhincfun 49098 fullthinc 49099 fullthinc2 49100 thincfth 49101 thincciso 49102

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