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Theorem xpchom 18128

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 483	. . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑢 = 𝑋)
4	3	fveq2d 6892	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1^st ‘𝑢) = (1^st ‘𝑋))
5		simpr 485	. . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑣 = 𝑌)
6	5	fveq2d 6892	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1^st ‘𝑣) = (1^st ‘𝑌))
7	4, 6	oveq12d 7423	. . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((1^st ‘𝑢)𝐻(1^st ‘𝑣)) = ((1^st ‘𝑋)𝐻(1^st ‘𝑌)))
8	3	fveq2d 6892	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2^nd ‘𝑢) = (2^nd ‘𝑋))
9	5	fveq2d 6892	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2^nd ‘𝑣) = (2^nd ‘𝑌))
10	8, 9	oveq12d 7423	. . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)) = ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌)))
11	7, 10	xpeq12d 5706	. . 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 18127	. . 3 ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))
18		ovex 7438	. . . 4 ⊢ ((1^st ‘𝑋)𝐻(1^st ‘𝑌)) ∈ V
19		ovex 7438	. . . 4 ⊢ ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌)) ∈ V
20	18, 19	xpex 7736	. . 3 ⊢ (((1^st ‘𝑋)𝐻(1^st ‘𝑌)) × ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌))) ∈ V
21	11, 17, 20	ovmpoa 7559	. 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 584	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 396 = wceq 1541 ∈ wcel 2106 × cxp 5673 ‘cfv 6540 (class class class)co 7405 1^st c1st 7969 2^nd c2nd 7970 Basecbs 17140 Hom chom 17204 ×_c cxpc 18116

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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-hom 17217 df-cco 17218 df-xpc 18120

This theorem is referenced by: xpchom2 18134 xpccatid 18136 1stfcl 18145 2ndfcl 18146 xpcpropd 18157 evlfcl 18171 curf1cl 18177 hofcl 18208 yonedalem3 18229

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