funcf2 - Metamath Proof Explorer

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Theorem funcf2 18048

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 7423	. . . 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 18047	. . . . 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 5689	. . . . 5 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
10		2fveq3 6890	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(1^st ‘𝑧)) = (𝐹‘(1^st ‘⟨𝑋, 𝑌⟩)))
11		2fveq3 6890	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐹‘(2^nd ‘𝑧)) = (𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩)))
12	10, 11	oveq12d 7438	. . . . . . 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 6885	. . . . . . . 8 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻‘𝑧) = (𝐻‘⟨𝑋, 𝑌⟩))
14		df-ov 7423	. . . . . . . 8 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
15	13, 14	eqtr4di 2813	. . . . . . 7 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻‘𝑧) = (𝑋𝐻𝑌))
16	12, 15	oveq12d 7438	. . . . . 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 8933	. . . . 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 596	. . . 4 ⊢ (𝜑 → (𝐺‘⟨𝑋, 𝑌⟩) ∈ (((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) ↑_m (𝑋𝐻𝑌)))
19	1, 18	eqeltrid 2864	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) ↑_m (𝑋𝐻𝑌)))
20		op1stg 8013	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
21	20	fveq2d 6889	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(1^st ‘⟨𝑋, 𝑌⟩)) = (𝐹‘𝑋))
22		op2ndg 8014	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
23	22	fveq2d 6889	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩)) = (𝐹‘𝑌))
24	21, 23	oveq12d 7438	. . . . 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 596	. . . 4 ⊢ (𝜑 → ((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
26	25	oveq1d 7435	. . 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 2862	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↑_m (𝑋𝐻𝑌)))
28		elmapi 8872	. 2 ⊢ ((𝑋𝐺𝑌) ∈ (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↑_m (𝑋𝐻𝑌)) → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
29	27, 28	syl 18	1 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ⟨cop 4834 class class class wbr 5102 × cxp 5648 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 1^st c1st 7999 2^nd c2nd 8000 ↑_m cmap 8850 Xcixp 8928 Basecbs 17392 Hom chom 17444 Func cfunc 18034

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 8001 df-2nd 8002 df-map 8852 df-ixp 8929 df-func 18038

This theorem is used by: funcsect 18052 funcoppc 18055 cofu2 18066 cofucl 18068 cofulid 18070 cofurid 18071 funcres 18076 funcres2 18078 funcres2c 18083 isfull2 18093 isfth2 18097 fthsect 18107 fthmon 18109 fuccocl 18147 fucidcl 18148 invfuc 18157 natpropd 18159 catciso 18291 prfval 18378 prfcl 18382 prf1st 18383 prf2nd 18384 1st2ndprf 18385 evlfcllem 18400 evlfcl 18401 curf1cl 18407 curf2cl 18410 uncf2 18416 curfuncf 18417 uncfcurf 18418 diag2cl 18425 curf2ndf 18426 yonedalem4c 18456 yonedalem3b 18458 yonedainv 18460 yonffthlem 18461 funchomf 50306 cofidf2a 50326 imassc 50362 imaid 50363 imaf1co 50364 upciclem2 50376 upeu2 50381 uppropd 50390 uptrlem1 50419 uptrlem3 50421 diag1 50519 diag2f1 50524 fuco112xa 50548 fuco22natlem1 50557 fuco22natlem2 50558 fuco22natlem3 50559 fuco22natlem 50560 fucocolem1 50568 fucocolem3 50570 fucoco 50572 fucolid 50576 prcofdiag1 50621 prcofdiag 50622 oppfdiag1 50642 oppfdiag 50644 functhincfun 50677 fullthinc 50678 fullthinc2 50679 thincfth 50680 thincciso 50681 termcfuncval 50760

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