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Theorem xpchom 18142

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 482	. . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑢 = 𝑋)
4	3	fveq2d 6895	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1^st ‘𝑢) = (1^st ‘𝑋))
5		simpr 484	. . . . . 6 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → 𝑣 = 𝑌)
6	5	fveq2d 6895	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (1^st ‘𝑣) = (1^st ‘𝑌))
7	4, 6	oveq12d 7430	. . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((1^st ‘𝑢)𝐻(1^st ‘𝑣)) = ((1^st ‘𝑋)𝐻(1^st ‘𝑌)))
8	3	fveq2d 6895	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2^nd ‘𝑢) = (2^nd ‘𝑋))
9	5	fveq2d 6895	. . . . 5 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → (2^nd ‘𝑣) = (2^nd ‘𝑌))
10	8, 9	oveq12d 7430	. . . 4 ⊢ ((𝑢 = 𝑋 ∧ 𝑣 = 𝑌) → ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)) = ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌)))
11	7, 10	xpeq12d 5707	. . 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 18141	. . 3 ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))
18		ovex 7445	. . . 4 ⊢ ((1^st ‘𝑋)𝐻(1^st ‘𝑌)) ∈ V
19		ovex 7445	. . . 4 ⊢ ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌)) ∈ V
20	18, 19	xpex 7744	. . 3 ⊢ (((1^st ‘𝑋)𝐻(1^st ‘𝑌)) × ((2^nd ‘𝑋)𝐽(2^nd ‘𝑌))) ∈ V
21	11, 17, 20	ovmpoa 7566	. 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 583	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 395 = wceq 1540 ∈ wcel 2105 × cxp 5674 ‘cfv 6543 (class class class)co 7412 1^st c1st 7977 2^nd c2nd 7978 Basecbs 17151 Hom chom 17215 ×_c cxpc 18130

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-hom 17228 df-cco 17229 df-xpc 18134

This theorem is referenced by: xpchom2 18148 xpccatid 18150 1stfcl 18159 2ndfcl 18160 xpcpropd 18171 evlfcl 18185 curf1cl 18191 hofcl 18222 yonedalem3 18243

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