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 18118

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 5650	. . . 4 ⊢ (((1^st ‘𝑢)(Hom ‘𝐶)(1^st ‘𝑣)) × ((2^nd ‘𝑢)(Hom ‘𝐷)(2^nd ‘𝑣))) ⊆ (V × V)
2	1	rgen2w 3057	. . 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 2737	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
5		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
6		eqid 2737	. . . . 5 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7		relxpchom.k	. . . . 5 ⊢ 𝐾 = (Hom ‘𝑇)
8	3, 4, 5, 6, 7	xpchomfval 18116	. . . 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 8047	. . 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 5641	. 2 ⊢ (Rel (𝑋𝐾𝑌) ↔ (𝑋𝐾𝑌) ⊆ (V × V))
12	10, 11	mpbir 231	1 ⊢ Rel (𝑋𝐾𝑌)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∀wral 3052 Vcvv 3442 ⊆ wss 3903 × cxp 5632 Rel wrel 5639 ‘cfv 6502 (class class class)co 7370 1^st c1st 7943 2^nd c2nd 7944 Basecbs 17150 Hom chom 17202 ×_c cxpc 18105

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-er 8647 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-fz 13438 df-struct 17088 df-slot 17123 df-ndx 17135 df-base 17151 df-hom 17215 df-cco 17216 df-xpc 18109

This theorem is referenced by: 1st2ndprf 18143

W3C validator