Theorem oppcbasOLD 17697

Description: Obsolete version of oppcbas 17696 as of 18-Oct-2024. Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
oppcbas.1	⊢ 𝑂 = (oppCat‘𝐶)
oppcbas.2	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
oppcbasOLD	⊢ 𝐵 = (Base‘𝑂)

Proof of Theorem oppcbasOLD

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

Step	Hyp	Ref	Expression
1		oppcbas.2	. 2 ⊢ 𝐵 = (Base‘𝐶)
2		baseid 17180	. . . . . 6 ⊢ Base = Slot (Base‘ndx)
3		1re 11242	. . . . . . . 8 ⊢ 1 ∈ ℝ
4		1nn 12251	. . . . . . . . 9 ⊢ 1 ∈ ℕ
5		4nn0 12519	. . . . . . . . 9 ⊢ 4 ∈ ℕ0
6		1nn0 12516	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
7		1lt10 12844	. . . . . . . . 9 ⊢ 1 < ;10
8	4, 5, 6, 7	declti 12743	. . . . . . . 8 ⊢ 1 < ;14
9	3, 8	ltneii 11355	. . . . . . 7 ⊢ 1 ≠ ;14
10		basendx 17186	. . . . . . . 8 ⊢ (Base‘ndx) = 1
11		homndx 17389	. . . . . . . 8 ⊢ (Hom ‘ndx) = ;14
12	10, 11	neeq12i 2997	. . . . . . 7 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14)
13	9, 12	mpbir 230	. . . . . 6 ⊢ (Base‘ndx) ≠ (Hom ‘ndx)
14	2, 13	setsnid 17175	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩))
15		5nn 12326	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
16		4lt5 12417	. . . . . . . . . 10 ⊢ 4 < 5
17	6, 5, 15, 16	declt 12733	. . . . . . . . 9 ⊢ ;14 < ;15
18		4nn 12323	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ
19	6, 18	decnncl 12725	. . . . . . . . . . 11 ⊢ ;14 ∈ ℕ
20	19	nnrei 12249	. . . . . . . . . 10 ⊢ ;14 ∈ ℝ
21	6, 15	decnncl 12725	. . . . . . . . . . 11 ⊢ ;15 ∈ ℕ
22	21	nnrei 12249	. . . . . . . . . 10 ⊢ ;15 ∈ ℝ
23	3, 20, 22	lttri 11368	. . . . . . . . 9 ⊢ ((1 < ;14 ∧ ;14 < ;15) → 1 < ;15)
24	8, 17, 23	mp2an 690	. . . . . . . 8 ⊢ 1 < ;15
25	3, 24	ltneii 11355	. . . . . . 7 ⊢ 1 ≠ ;15
26		ccondx 17391	. . . . . . . 8 ⊢ (comp‘ndx) = ;15
27	10, 26	neeq12i 2997	. . . . . . 7 ⊢ ((Base‘ndx) ≠ (comp‘ndx) ↔ 1 ≠ ;15)
28	25, 27	mpbir 230	. . . . . 6 ⊢ (Base‘ndx) ≠ (comp‘ndx)
29	2, 28	setsnid 17175	. . . . 5 ⊢ (Base‘(𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩)) = (Base‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑢)))⟩))
30	14, 29	eqtri 2753	. . . 4 ⊢ (Base‘𝐶) = (Base‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑢)))⟩))
31		eqid 2725	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
32		eqid 2725	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
33		eqid 2725	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
34		oppcbas.1	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
35	31, 32, 33, 34	oppcval 17690	. . . . 5 ⊢ (𝐶 ∈ V → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑢)))⟩))
36	35	fveq2d 6895	. . . 4 ⊢ (𝐶 ∈ V → (Base‘𝑂) = (Base‘((𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑢)))⟩)))
37	30, 36	eqtr4id 2784	. . 3 ⊢ (𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
38		base0 17182	. . . 4 ⊢ ∅ = (Base‘∅)
39		fvprc 6883	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
40		fvprc 6883	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅)
41	34, 40	eqtrid 2777	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅)
42	41	fveq2d 6895	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝑂) = (Base‘∅))
43	38, 39, 42	3eqtr4a 2791	. . 3 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
44	37, 43	pm2.61i 182	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
45	1, 44	eqtri 2753	1 ⊢ 𝐵 = (Base‘𝑂)

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  Vcvv 3463  ∅c0 4318  ⟨cop 4630   class class class wbr 5143   × cxp 5670  ‘cfv 6542  (class class class)co 7415   ∈ cmpo 7417  1^st c1st 7987  2^nd c2nd 7988  tpos ctpos 8227  1c1 11137   < clt 11276  4c4 12297  5c5 12298  ;cdc 12705   sSet csts 17129  ndxcnx 17159  Basecbs 17177  Hom chom 17241  compcco 17242  oppCatcoppc 17688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-2nd 7990  df-tpos 8228  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12501  df-z 12587  df-dec 12706  df-sets 17130  df-slot 17148  df-ndx 17160  df-base 17178  df-hom 17254  df-cco 17255  df-oppc 17689

This theorem is referenced by: (None)