Theorem oppchomfvalOLD 17341

Description: Obsolete proof of oppchomfval 17340 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 17041	. . . 4 ⊢ Hom = Slot (Hom ‘ndx)
2		1nn0 12179	. . . . . . . 8 ⊢ 1 ∈ ℕ0
3		4nn 11986	. . . . . . . 8 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12386	. . . . . . 7 ⊢ ;14 ∈ ℕ
5	4	nnrei 11912	. . . . . 6 ⊢ ;14 ∈ ℝ
6		4nn0 12182	. . . . . . 7 ⊢ 4 ∈ ℕ0
7		5nn 11989	. . . . . . 7 ⊢ 5 ∈ ℕ
8		4lt5 12080	. . . . . . 7 ⊢ 4 < 5
9	2, 6, 7, 8	declt 12394	. . . . . 6 ⊢ ;14 < ;15
10	5, 9	ltneii 11018	. . . . 5 ⊢ ;14 ≠ ;15
11		homndx 17040	. . . . . 6 ⊢ (Hom ‘ndx) = ;14
12		ccondx 17042	. . . . . 6 ⊢ (comp‘ndx) = ;15
13	11, 12	neeq12i 3009	. . . . 5 ⊢ ((Hom ‘ndx) ≠ (comp‘ndx) ↔ ;14 ≠ ;15)
14	10, 13	mpbir 230	. . . 4 ⊢ (Hom ‘ndx) ≠ (comp‘ndx)
15	1, 14	setsnid 16838	. . 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 6770	. . . . 5 ⊢ 𝐻 ∈ V
18	17	tposex 8047	. . . 4 ⊢ tpos 𝐻 ∈ V
19	1	setsid 16837	. . . 4 ⊢ ((𝐶 ∈ V ∧ tpos 𝐻 ∈ V) → tpos 𝐻 = (Hom ‘(𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩)))
20	18, 19	mpan2 687	. . 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 17339	. . . 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 6760	. . 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 2805	. 2 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂))
27		tpos0 8043	. . 3 ⊢ tpos ∅ = ∅
28		fvprc 6748	. . . . 5 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝐶) = ∅)
29	16, 28	eqtrid 2790	. . . 4 ⊢ (¬ 𝐶 ∈ V → 𝐻 = ∅)
30	29	tposeqd 8016	. . 3 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = tpos ∅)
31		fvprc 6748	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅)
32	23, 31	eqtrid 2790	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅)
33	32	fveq2d 6760	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = (Hom ‘∅))
34		df-hom 16912	. . . . 5 ⊢ Hom = Slot ;14
35	34	str0 16818	. . . 4 ⊢ ∅ = (Hom ‘∅)
36	33, 35	eqtr4di 2797	. . 3 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = ∅)
37	27, 30, 36	3eqtr4a 2805	. 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 2108   ≠ wne 2942  Vcvv 3422  ∅c0 4253  ⟨cop 4564   × cxp 5578  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  1^st c1st 7802  2^nd c2nd 7803  tpos ctpos 8012  1c1 10803  4c4 11960  5c5 11961  ;cdc 12366   sSet csts 16792  ndxcnx 16822  Basecbs 16840  Hom chom 16899  compcco 16900  oppCatcoppc 17337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-tpos 8013  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-dec 12367  df-sets 16793  df-slot 16811  df-ndx 16823  df-hom 16912  df-cco 16913  df-oppc 17338

This theorem is referenced by: (None)