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Theorem xpchomfval 18333

Description: Set of morphisms of the binary product of categories. (Contributed by Mario Carneiro, 11-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)

**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)

**Assertion**
Ref	Expression
xpchomfval	⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))

Distinct variable groups: 𝑣,𝑢,𝐵 𝑢,𝐶,𝑣 𝑢,𝐷,𝑣 𝑢,𝐻,𝑣 𝑢,𝐽,𝑣 Allowed substitution hints: 𝑇(𝑣, 𝑢) 𝐾(𝑣, 𝑢)

Proof of Theorem xpchomfval Dummy variables 𝑓 𝑔 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpchomfval.t	. . . 4 ⊢ 𝑇 = (𝐶 ×_c 𝐷)
2		eqid 2761	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2761	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
4		xpchomfval.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
5		xpchomfval.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐷)
6		eqid 2761	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
7		eqid 2761	. . . 4 ⊢ (comp‘𝐷) = (comp‘𝐷)
8		simpl 488	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V)
9		simpr 490	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V)
10		xpchomfval.y	. . . . . 6 ⊢ 𝐵 = (Base‘𝑇)
11	1, 2, 3	xpcbas 18332	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
12	10, 11	eqtr4i 2787	. . . . 5 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
13	12	a1i 11	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ((Base‘𝐶) × (Base‘𝐷)))
14		eqidd 2762	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))) = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))))
15		eqidd 2762	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)(𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)(𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩)))
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 15	xpcval 18331	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)(𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩})
17		catstr 18115	. . 3 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)(𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩} Struct ⟨1, ;15⟩
18		homid 17563	. . 3 ⊢ Hom = Slot (Hom ‘ndx)
19		snsstp2 4778	. . 3 ⊢ {⟨(Hom ‘ndx), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))⟩, ⟨(comp‘ndx), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2^nd ‘𝑥)(𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))‘𝑥) ↦ ⟨((1^st ‘𝑔)(⟨(1^st ‘(1^st ‘𝑥)), (1^st ‘(2^nd ‘𝑥))⟩(comp‘𝐶)(1^st ‘𝑦))(1^st ‘𝑓)), ((2^nd ‘𝑔)(⟨(2^nd ‘(1^st ‘𝑥)), (2^nd ‘(2^nd ‘𝑥))⟩(comp‘𝐷)(2^nd ‘𝑦))(2^nd ‘𝑓))⟩))⟩}
20	10	fvexi 6891	. . . . 5 ⊢ 𝐵 ∈ V
21	20, 20	mpoex 8081	. . . 4 ⊢ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))) ∈ V
22	21	a1i 11	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))) ∈ V)
23		xpchomfval.k	. . 3 ⊢ 𝐾 = (Hom ‘𝑇)
24	16, 17, 18, 19, 22, 23	strfv3 17362	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))))
25		fnxpc 18330	. . . . . . . 8 ⊢ ×_c Fn (V × V)
26		fndm 6634	. . . . . . . 8 ⊢ ( ×_c Fn (V × V) → dom ×_c = (V × V))
27	25, 26	ax-mp 5	. . . . . . 7 ⊢ dom ×_c = (V × V)
28	27	ndmov 7597	. . . . . 6 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×_c 𝐷) = ∅)
29	1, 28	eqtrid 2808	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅)
30	29	fveq2d 6881	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Hom ‘𝑇) = (Hom ‘∅))
31	18	str0 17347	. . . 4 ⊢ ∅ = (Hom ‘∅)
32	30, 23, 31	3eqtr4g 2821	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = ∅)
33	29	fveq2d 6881	. . . . . 6 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (Base‘𝑇) = (Base‘∅))
34		base0 17372	. . . . . 6 ⊢ ∅ = (Base‘∅)
35	33, 10, 34	3eqtr4g 2821	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐵 = ∅)
36	35	olcd 888	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐵 = ∅ ∨ 𝐵 = ∅))
37		0mpo0 7495	. . . 4 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))) = ∅)
38	36, 37	syl 18	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))) = ∅)
39	32, 38	eqtr4d 2799	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣)))))
40	24, 39	pm2.61i 184	1 ⊢ 𝐾 = (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (((1^st ‘𝑢)𝐻(1^st ‘𝑣)) × ((2^nd ‘𝑢)𝐽(2^nd ‘𝑣))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 Vcvv 3451 ∅c0 4279 {ctp 4588 ⟨cop 4590 × cxp 5649 dom cdm 5651 Fn wfn 6526 ‘cfv 6531 (class class class)co 7412 ∈ cmpo 7414 1^st c1st 7988 2^nd c2nd 7989 1c1 11182 5c5 12381 cdc 12795 ndxcnx 17351 Basecbs 17367 Hom chom 17419 compcco 17420 ×_c cxpc 18322

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-er 8701 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-z 12675 df-dec 12796 df-uz 12947 df-fz 13621 df-struct 17305 df-slot 17340 df-ndx 17352 df-base 17368 df-hom 17432 df-cco 17433 df-xpc 18326

This theorem is used by: xpchom 18334 relxpchom 18335 xpccofval 18336 catcxpccl 18361 xpcpropd 18362 xpcfuchomfval 50305

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