relxpchom - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > relxpchom		Structured version Visualization version GIF version

Theorem relxpchom 16868

Description: A hom-set in the binary product of categories is a relation. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
relxpchom.t	⊢ 𝑇 = (𝐶 ×_c 𝐷)
relxpchom.k	⊢ 𝐾 = (Hom ‘𝑇)

**Assertion**
Ref	Expression
relxpchom	⊢ Rel (𝑋𝐾𝑌)

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

Step	Hyp	Ref	Expression
1		xpss 5159	. . . 4 ⊢ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))) ⊆ (V × V)
2	1	rgen2w 2954	. . 3 ⊢ ∀𝑢 ∈ (Base‘𝑇)∀𝑣 ∈ (Base‘𝑇)(((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))) ⊆ (V × V)
3		relxpchom.t	. . . . 5 ⊢ 𝑇 = (𝐶 ×_c 𝐷)
4		eqid 2651	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
5		eqid 2651	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2651	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7		relxpchom.k	. . . . 5 ⊢ 𝐾 = (Hom ‘𝑇)
8	3, 4, 5, 6, 7	xpchomfval 16866	. . . 4 ⊢ 𝐾 = (𝑢 ∈ (Base‘𝑇), 𝑣 ∈ (Base‘𝑇) ↦ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))))
9	8	ovmptss 7303	. . 3 ⊢ (∀𝑢 ∈ (Base‘𝑇)∀𝑣 ∈ (Base‘𝑇)(((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))) ⊆ (V × V) → (𝑋𝐾𝑌) ⊆ (V × V))
10	2, 9	ax-mp 5	. 2 ⊢ (𝑋𝐾𝑌) ⊆ (V × V)
11		df-rel 5150	. 2 ⊢ (Rel (𝑋𝐾𝑌) ↔ (𝑋𝐾𝑌) ⊆ (V × V))
12	10, 11	mpbir 221	1 ⊢ Rel (𝑋𝐾𝑌)

Colors of variables: wff setvar class

Syntax hints: = wceq 1523 ∀wral 2941 Vcvv 3231 ⊆ wss 3607 × cxp 5141 Rel wrel 5148 ‘cfv 5926 (class class class)co 6690 1^st c1st 7208 2^nd c2nd 7209 Basecbs 15904 Hom chom 15999 ×_c cxpc 16855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-hom 16013 df-cco 16014 df-xpc 16859

This theorem is referenced by: 1st2ndprf 16893

W3C validator