funcf2 - Metamath Proof Explorer

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Theorem funcf2 17775

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 7352	. . . 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 17774	. . . . 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 5658	. . . . 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 7367	. . . . . . 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 7352	. . . . . . . 8 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
15	13, 14	eqtr4di 2782	. . . . . . 7 ⊢ (𝑧 = ⟨𝑋, 𝑌⟩ → (𝐻‘𝑧) = (𝑋𝐻𝑌))
16	12, 15	oveq12d 7367	. . . . . 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 8829	. . . . 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 2832	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘(1^st ‘⟨𝑋, 𝑌⟩))𝐽(𝐹‘(2^nd ‘⟨𝑋, 𝑌⟩))) ↑_m (𝑋𝐻𝑌)))
20		op1stg 7936	. . . . . . 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 7937	. . . . . . 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 7367	. . . . 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 7364	. . 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 2830	. 2 ⊢ (𝜑 → (𝑋𝐺𝑌) ∈ (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) ↑_m (𝑋𝐻𝑌)))
28		elmapi 8776	. 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 2109 ⟨cop 4583 class class class wbr 5092 × cxp 5617 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 1^st c1st 7922 2^nd c2nd 7923 ↑_m cmap 8753 Xcixp 8824 Basecbs 17120 Hom chom 17172 Func cfunc 17761

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7924 df-2nd 7925 df-map 8755 df-ixp 8825 df-func 17765

This theorem is referenced by: funcsect 17779 funcoppc 17782 cofu2 17793 cofucl 17795 cofulid 17797 cofurid 17798 funcres 17803 funcres2 17805 funcres2c 17810 isfull2 17820 isfth2 17824 fthsect 17834 fthmon 17836 fuccocl 17874 fucidcl 17875 invfuc 17884 natpropd 17886 catciso 18018 prfval 18105 prfcl 18109 prf1st 18110 prf2nd 18111 1st2ndprf 18112 evlfcllem 18127 evlfcl 18128 curf1cl 18134 curf2cl 18137 uncf2 18143 curfuncf 18144 uncfcurf 18145 diag2cl 18152 curf2ndf 18153 yonedalem4c 18183 yonedalem3b 18185 yonedainv 18187 yonffthlem 18188 funchomf 49092 cofidf2a 49112 imassc 49148 imaid 49149 imaf1co 49150 upciclem2 49162 upeu2 49167 uppropd 49176 uptrlem1 49205 uptrlem3 49207 diag1 49299 diag2f1 49304 fuco112xa 49328 fuco22natlem1 49337 fuco22natlem2 49338 fuco22natlem3 49339 fuco22natlem 49340 fucocolem1 49348 fucocolem3 49350 fucoco 49352 fucolid 49356 prcofdiag1 49388 prcofdiag 49389 oppfdiag1 49409 oppfdiag 49411 functhincfun 49444 fullthinc 49445 fullthinc2 49446 thincfth 49447 thincciso 49448 termcfuncval 49527

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