relxpchom - Metamath Proof Explorer

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Theorem relxpchom 18207

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 5700	. . . 4 ⊢ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))) ⊆ (V × V)
2	1	rgen2w 3056	. . 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 2726	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
5		eqid 2726	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2726	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7		relxpchom.k	. . . . 5 ⊢ 𝐾 = (Hom ‘𝑇)
8	3, 4, 5, 6, 7	xpchomfval 18205	. . . 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 8109	. . 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 5691	. 2 ⊢ (Rel (𝑋𝐾𝑌) ↔ (𝑋𝐾𝑌) ⊆ (V × V))
12	10, 11	mpbir 230	1 ⊢ Rel (𝑋𝐾𝑌)

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∀wral 3051 Vcvv 3462 ⊆ wss 3947 × cxp 5682 Rel wrel 5689 ‘cfv 6556 (class class class)co 7426 1^st c1st 8003 2^nd c2nd 8004 Basecbs 17215 Hom chom 17279 ×_c cxpc 18194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-1st 8005 df-2nd 8006 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-er 8736 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12613 df-dec 12732 df-uz 12877 df-fz 13541 df-struct 17151 df-slot 17186 df-ndx 17198 df-base 17216 df-hom 17292 df-cco 17293 df-xpc 18198

This theorem is referenced by: 1st2ndprf 18232

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