relxpchom - Metamath Proof Explorer

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Theorem relxpchom 18066

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 5648	. . . 4 ⊢ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))) ⊆ (V × V)
2	1	rgen2w 3068	. . 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 2736	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
5		eqid 2736	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2736	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7		relxpchom.k	. . . . 5 ⊢ 𝐾 = (Hom ‘𝑇)
8	3, 4, 5, 6, 7	xpchomfval 18064	. . . 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 8022	. . 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 5639	. 2 ⊢ (Rel (𝑋𝐾𝑌) ↔ (𝑋𝐾𝑌) ⊆ (V × V))
12	10, 11	mpbir 230	1 ⊢ Rel (𝑋𝐾𝑌)

Colors of variables: wff setvar class

Syntax hints: = wceq 1541 ∀wral 3063 Vcvv 3444 ⊆ wss 3909 × cxp 5630 Rel wrel 5637 ‘cfv 6494 (class class class)co 7354 1^st c1st 7916 2^nd c2nd 7917 Basecbs 17080 Hom chom 17141 ×_c cxpc 18053

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12411 df-z 12497 df-dec 12616 df-uz 12761 df-fz 13422 df-struct 17016 df-slot 17051 df-ndx 17063 df-base 17081 df-hom 17154 df-cco 17155 df-xpc 18057

This theorem is referenced by: 1st2ndprf 18091

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