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Theorem fuchomOLD 17916

Description: Obsolete proof of fuchom 17915 as of 14-Oct-2024. (Contributed by Mario Carneiro, 6-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
fucbas.q	⊢ 𝑄 = (𝐶 FuncCat 𝐷)
fuchom.n	⊢ 𝑁 = (𝐶 Nat 𝐷)

**Assertion**
Ref	Expression
fuchomOLD	⊢ 𝑁 = (Hom ‘𝑄)

Proof of Theorem fuchomOLD Dummy variables 𝑎 𝑏 𝑓 𝑔 ℎ 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fucbas.q	. . . . 5 ⊢ 𝑄 = (𝐶 FuncCat 𝐷)
2		eqid 2732	. . . . 5 ⊢ (𝐶 Func 𝐷) = (𝐶 Func 𝐷)
3		fuchom.n	. . . . 5 ⊢ 𝑁 = (𝐶 Nat 𝐷)
4		eqid 2732	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2732	. . . . 5 ⊢ (comp‘𝐷) = (comp‘𝐷)
6		simpl 483	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐶 ∈ Cat)
7		simpr 485	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat)
8		eqid 2732	. . . . . 6 ⊢ (comp‘𝑄) = (comp‘𝑄)
9	1, 2, 3, 4, 5, 6, 7, 8	fuccofval 17913	. . . . 5 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (comp‘𝑄) = (𝑣 ∈ ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)), ℎ ∈ (𝐶 Func 𝐷) ↦ ⦋(1^st ‘𝑣) / 𝑓⦌⦋(2^nd ‘𝑣) / 𝑔⦌(𝑏 ∈ (𝑔𝑁ℎ), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘𝑥)(⟨((1^st ‘𝑓)‘𝑥), ((1^st ‘𝑔)‘𝑥)⟩(comp‘𝐷)((1^st ‘ℎ)‘𝑥))(𝑎‘𝑥))))))
10	1, 2, 3, 4, 5, 6, 7, 9	fucval 17912	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = {⟨(Base‘ndx), (𝐶 Func 𝐷)⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (comp‘𝑄)⟩})
11		catstr 17911	. . . 4 ⊢ {⟨(Base‘ndx), (𝐶 Func 𝐷)⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (comp‘𝑄)⟩} Struct ⟨1, ;15⟩
12		homid 17359	. . . 4 ⊢ Hom = Slot (Hom ‘ndx)
13		snsstp2 4820	. . . 4 ⊢ {⟨(Hom ‘ndx), 𝑁⟩} ⊆ {⟨(Base‘ndx), (𝐶 Func 𝐷)⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), (comp‘𝑄)⟩}
14	3	ovexi 7445	. . . . 5 ⊢ 𝑁 ∈ V
15	14	a1i 11	. . . 4 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 ∈ V)
16		eqid 2732	. . . 4 ⊢ (Hom ‘𝑄) = (Hom ‘𝑄)
17	10, 11, 12, 13, 15, 16	strfv3 17140	. . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = 𝑁)
18	17	eqcomd 2738	. 2 ⊢ ((𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = (Hom ‘𝑄))
19		df-hom 17223	. . . 4 ⊢ Hom = Slot ;14
20	19	str0 17124	. . 3 ⊢ ∅ = (Hom ‘∅)
21	3	natffn 17902	. . . . 5 ⊢ 𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷))
22		funcrcl 17815	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
23	22	con3i 154	. . . . . . . . 9 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ¬ 𝑓 ∈ (𝐶 Func 𝐷))
24	23	eq0rdv 4404	. . . . . . . 8 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 Func 𝐷) = ∅)
25	24	xpeq2d 5706	. . . . . . 7 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ((𝐶 Func 𝐷) × ∅))
26		xp0 6157	. . . . . . 7 ⊢ ((𝐶 Func 𝐷) × ∅) = ∅
27	25, 26	eqtrdi 2788	. . . . . 6 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) = ∅)
28	27	fneq2d 6643	. . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝑁 Fn ((𝐶 Func 𝐷) × (𝐶 Func 𝐷)) ↔ 𝑁 Fn ∅))
29	21, 28	mpbii 232	. . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 Fn ∅)
30		fn0 6681	. . . 4 ⊢ (𝑁 Fn ∅ ↔ 𝑁 = ∅)
31	29, 30	sylib 217	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = ∅)
32		fnfuc 17898	. . . . . . 7 ⊢ FuncCat Fn (Cat × Cat)
33	32	fndmi 6653	. . . . . 6 ⊢ dom FuncCat = (Cat × Cat)
34	33	ndmov 7593	. . . . 5 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (𝐶 FuncCat 𝐷) = ∅)
35	1, 34	eqtrid 2784	. . . 4 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑄 = ∅)
36	35	fveq2d 6895	. . 3 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → (Hom ‘𝑄) = (Hom ‘∅))
37	20, 31, 36	3eqtr4a 2798	. 2 ⊢ (¬ (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat) → 𝑁 = (Hom ‘𝑄))
38	18, 37	pm2.61i 182	1 ⊢ 𝑁 = (Hom ‘𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4322 {ctp 4632 ⟨cop 4634 × cxp 5674 Fn wfn 6538 ‘cfv 6543 (class class class)co 7411 1c1 11113 4c4 12271 5c5 12272 cdc 12679 ndxcnx 17128 Basecbs 17146 Hom chom 17210 compcco 17211 Catccat 17610 Func cfunc 17806 Nat cnat 17894 FuncCat cfuc 17895

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

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 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-fz 13487 df-struct 17082 df-slot 17117 df-ndx 17129 df-base 17147 df-hom 17223 df-cco 17224 df-func 17810 df-nat 17896 df-fuc 17897

This theorem is referenced by: (None)

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