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Theorem xpchom 16820

Description: Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
xpchomfval.t	⊢ 𝑇 = (𝐶 ×_c 𝐷)
xpchomfval.y	⊢ 𝐵 = (Base‘𝑇)
xpchomfval.h	⊢ 𝐻 = (Hom ‘𝐶)
xpchomfval.j	⊢ 𝐽 = (Hom ‘𝐷)
xpchomfval.k	⊢ 𝐾 = (Hom ‘𝑇)
xpchom.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
xpchom.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
xpchom	⊢ (𝜑 → (𝑋𝐾𝑌) = (((1^st ‘𝑋)𝐻(1^st ‘𝑌)) × ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌))))

Proof of Theorem xpchom Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpchom.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		xpchom.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
3		simpl 473	. . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑢 = 𝑋)
4	3	fveq2d 6195	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1^st ‘𝑢) = (1^st ‘𝑋))
5		simpr 477	. . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑣 = 𝑌)
6	5	fveq2d 6195	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1^st ‘𝑣) = (1^st ‘𝑌))
7	4, 6	oveq12d 6668	. . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((1^st ‘𝑢)𝐻(1^st ‘𝑣)) = ((1^st ‘𝑋)𝐻(1^st ‘𝑌)))
8	3	fveq2d 6195	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2^nd ‘𝑢) = (2^nd ‘𝑋))
9	5	fveq2d 6195	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2^nd ‘𝑣) = (2^nd ‘𝑌))
10	8, 9	oveq12d 6668	. . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)) = ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌)))
11	7, 10	xpeq12d 5140	. . 3 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))) = (((1^st ‘𝑋)𝐻(1^st ‘𝑌)) × ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌))))
12		xpchomfval.t	. . . 4 ⊢ 𝑇 = (𝐶 ×_c 𝐷)
13		xpchomfval.y	. . . 4 ⊢ 𝐵 = (Base‘𝑇)
14		xpchomfval.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
15		xpchomfval.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐷)
16		xpchomfval.k	. . . 4 ⊢ 𝐾 = (Hom ‘𝑇)
17	12, 13, 14, 15, 16	xpchomfval 16819	. . 3 ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))
18		ovex 6678	. . . 4 ⊢ ((1^st ‘𝑋)𝐻(1^st ‘𝑌)) ∈ V
19		ovex 6678	. . . 4 ⊢ ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌)) ∈ V
20	18, 19	xpex 6962	. . 3 ⊢ (((1^st ‘𝑋)𝐻(1^st ‘𝑌)) × ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌))) ∈ V
21	11, 17, 20	ovmpt2a 6791	. 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐾𝑌) = (((1^st ‘𝑋)𝐻(1^st ‘𝑌)) × ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌))))
22	1, 2, 21	syl2anc 693	1 ⊢ (𝜑 → (𝑋𝐾𝑌) = (((1^st ‘𝑋)𝐻(1^st ‘𝑌)) × ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 × cxp 5112 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 Basecbs 15857 Hom chom 15952 ×_c cxpc 16808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-hom 15966 df-cco 15967 df-xpc 16812

This theorem is referenced by: xpchom2 16826 xpccatid 16828 1stfcl 16837 2ndfcl 16838 xpcpropd 16848 evlfcl 16862 curf1cl 16868 hofcl 16899 yonedalem3 16920

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