Theorem oppchomfvalOLD 17424

Description: Obsolete proof of oppchomfval 17423 as of 14-Oct-2024. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
oppchom.h	⊢ 𝐻 = (Hom ‘𝐶)
oppchom.o	⊢ 𝑂 = (oppCat‘𝐶)

Assertion

Ref	Expression
oppchomfvalOLD	⊢ tpos 𝐻 = (Hom ‘𝑂)

Proof of Theorem oppchomfvalOLD

Dummy variables 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		homid 17122	. . . 4 ⊢ Hom = Slot (Hom ‘ndx)
2		1nn0 12249	. . . . . . . 8 ⊢ 1 ∈ ℕ0
3		4nn 12056	. . . . . . . 8 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12457	. . . . . . 7 ⊢ ;14 ∈ ℕ
5	4	nnrei 11982	. . . . . 6 ⊢ ;14 ∈ ℝ
6		4nn0 12252	. . . . . . 7 ⊢ 4 ∈ ℕ0
7		5nn 12059	. . . . . . 7 ⊢ 5 ∈ ℕ
8		4lt5 12150	. . . . . . 7 ⊢ 4 < 5
9	2, 6, 7, 8	declt 12465	. . . . . 6 ⊢ ;14 < ;15
10	5, 9	ltneii 11088	. . . . 5 ⊢ ;14 ≠ ;15
11		homndx 17121	. . . . . 6 ⊢ (Hom ‘ndx) = ;14
12		ccondx 17123	. . . . . 6 ⊢ (comp‘ndx) = ;15
13	11, 12	neeq12i 3010	. . . . 5 ⊢ ((Hom ‘ndx) ≠ (comp‘ndx) ↔ ;14 ≠ ;15)
14	10, 13	mpbir 230	. . . 4 ⊢ (Hom ‘ndx) ≠ (comp‘ndx)
15	1, 14	setsnid 16910	. . 3 ⊢ (Hom ‘(𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩)) = (Hom ‘((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑢)))⟩))
16		oppchom.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
17	16	fvexi 6788	. . . . 5 ⊢ 𝐻 ∈ V
18	17	tposex 8076	. . . 4 ⊢ tpos 𝐻 ∈ V
19	1	setsid 16909	. . . 4 ⊢ ((𝐶 ∈ V ∧ tpos 𝐻 ∈ V) → tpos 𝐻 = (Hom ‘(𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩)))
20	18, 19	mpan2 688	. . 3 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘(𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩)))
21		eqid 2738	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		eqid 2738	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
23		oppchom.o	. . . . 5 ⊢ 𝑂 = (oppCat‘𝐶)
24	21, 16, 22, 23	oppcval 17422	. . . 4 ⊢ (𝐶 ∈ V → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑢)))⟩))
25	24	fveq2d 6778	. . 3 ⊢ (𝐶 ∈ V → (Hom ‘𝑂) = (Hom ‘((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑢)))⟩)))
26	15, 20, 25	3eqtr4a 2804	. 2 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂))
27		tpos0 8072	. . 3 ⊢ tpos ∅ = ∅
28		fvprc 6766	. . . . 5 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝐶) = ∅)
29	16, 28	eqtrid 2790	. . . 4 ⊢ (¬ 𝐶 ∈ V → 𝐻 = ∅)
30	29	tposeqd 8045	. . 3 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = tpos ∅)
31		fvprc 6766	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅)
32	23, 31	eqtrid 2790	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅)
33	32	fveq2d 6778	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = (Hom ‘∅))
34		df-hom 16986	. . . . 5 ⊢ Hom = Slot ;14
35	34	str0 16890	. . . 4 ⊢ ∅ = (Hom ‘∅)
36	33, 35	eqtr4di 2796	. . 3 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = ∅)
37	27, 30, 36	3eqtr4a 2804	. 2 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂))
38	26, 37	pm2.61i 182	1 ⊢ tpos 𝐻 = (Hom ‘𝑂)

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432  ∅c0 4256  ⟨cop 4567   × cxp 5587  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  1^st c1st 7829  2^nd c2nd 7830  tpos ctpos 8041  1c1 10872  4c4 12030  5c5 12031  ;cdc 12437   sSet csts 16864  ndxcnx 16894  Basecbs 16912  Hom chom 16973  compcco 16974  oppCatcoppc 17420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-dec 12438  df-sets 16865  df-slot 16883  df-ndx 16895  df-hom 16986  df-cco 16987  df-oppc 17421

This theorem is referenced by: (None)