Theorem oppchomfvalOLD 17172

Description: Obsolete proof of oppchomfval 17171 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 16873	. . . 4 ⊢ Hom = Slot (Hom ‘ndx)
2		1nn0 12071	. . . . . . . 8 ⊢ 1 ∈ ℕ0
3		4nn 11878	. . . . . . . 8 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 12278	. . . . . . 7 ⊢ ;14 ∈ ℕ
5	4	nnrei 11804	. . . . . 6 ⊢ ;14 ∈ ℝ
6		4nn0 12074	. . . . . . 7 ⊢ 4 ∈ ℕ0
7		5nn 11881	. . . . . . 7 ⊢ 5 ∈ ℕ
8		4lt5 11972	. . . . . . 7 ⊢ 4 < 5
9	2, 6, 7, 8	declt 12286	. . . . . 6 ⊢ ;14 < ;15
10	5, 9	ltneii 10910	. . . . 5 ⊢ ;14 ≠ ;15
11		homndx 16872	. . . . . 6 ⊢ (Hom ‘ndx) = ;14
12		ccondx 16874	. . . . . 6 ⊢ (comp‘ndx) = ;15
13	11, 12	neeq12i 2998	. . . . 5 ⊢ ((Hom ‘ndx) ≠ (comp‘ndx) ↔ ;14 ≠ ;15)
14	10, 13	mpbir 234	. . . 4 ⊢ (Hom ‘ndx) ≠ (comp‘ndx)
15	1, 14	setsnid 16720	. . 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 6709	. . . . 5 ⊢ 𝐻 ∈ V
18	17	tposex 7980	. . . 4 ⊢ tpos 𝐻 ∈ V
19	1	setsid 16719	. . . 4 ⊢ ((𝐶 ∈ V ∧ tpos 𝐻 ∈ V) → tpos 𝐻 = (Hom ‘(𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩)))
20	18, 19	mpan2 691	. . 3 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘(𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩)))
21		eqid 2736	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
22		eqid 2736	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
23		oppchom.o	. . . . 5 ⊢ 𝑂 = (oppCat‘𝐶)
24	21, 16, 22, 23	oppcval 17170	. . . 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 6699	. . 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 2797	. 2 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂))
27		tpos0 7976	. . 3 ⊢ tpos ∅ = ∅
28		fvprc 6687	. . . . 5 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝐶) = ∅)
29	16, 28	syl5eq 2783	. . . 4 ⊢ (¬ 𝐶 ∈ V → 𝐻 = ∅)
30	29	tposeqd 7949	. . 3 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = tpos ∅)
31		fvprc 6687	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅)
32	23, 31	syl5eq 2783	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅)
33	32	fveq2d 6699	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = (Hom ‘∅))
34		df-hom 16773	. . . . 5 ⊢ Hom = Slot ;14
35	34	str0 16717	. . . 4 ⊢ ∅ = (Hom ‘∅)
36	33, 35	eqtr4di 2789	. . 3 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = ∅)
37	27, 30, 36	3eqtr4a 2797	. 2 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂))
38	26, 37	pm2.61i 185	1 ⊢ tpos 𝐻 = (Hom ‘𝑂)

Syntax hints:  ¬ wn 3   = wceq 1543   ∈ wcel 2112   ≠ wne 2932  Vcvv 3398  ∅c0 4223  ⟨cop 4533   × cxp 5534  ‘cfv 6358  (class class class)co 7191   ∈ cmpo 7193  1^st c1st 7737  2^nd c2nd 7738  tpos ctpos 7945  1c1 10695  4c4 11852  5c5 11853  ;cdc 12258  ndxcnx 16663   sSet csts 16664  Basecbs 16666  Hom chom 16760  compcco 16761  oppCatcoppc 17168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-tpos 7946  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-nn 11796  df-2 11858  df-3 11859  df-4 11860  df-5 11861  df-6 11862  df-7 11863  df-8 11864  df-9 11865  df-n0 12056  df-dec 12259  df-ndx 16669  df-slot 16670  df-sets 16673  df-hom 16773  df-cco 16774  df-oppc 17169

This theorem is referenced by: (None)