Theorem oppcbasOLD 17726

Description: Obsolete version of oppcbas 17725 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 17209	. . . . . 6 ⊢ Base = Slot (Base‘ndx)
3		1re 11253	. . . . . . . 8 ⊢ 1 ∈ ℝ
4		1nn 12267	. . . . . . . . 9 ⊢ 1 ∈ ℕ
5		4nn0 12535	. . . . . . . . 9 ⊢ 4 ∈ ℕ0
6		1nn0 12532	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
7		1lt10 12860	. . . . . . . . 9 ⊢ 1 < ;10
8	4, 5, 6, 7	declti 12759	. . . . . . . 8 ⊢ 1 < ;14
9	3, 8	ltneii 11366	. . . . . . 7 ⊢ 1 ≠ ;14
10		basendx 17215	. . . . . . . 8 ⊢ (Base‘ndx) = 1
11		homndx 17418	. . . . . . . 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 17204	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩))
15		5nn 12342	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
16		4lt5 12433	. . . . . . . . . 10 ⊢ 4 < 5
17	6, 5, 15, 16	declt 12749	. . . . . . . . 9 ⊢ ;14 < ;15
18		4nn 12339	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ
19	6, 18	decnncl 12741	. . . . . . . . . . 11 ⊢ ;14 ∈ ℕ
20	19	nnrei 12265	. . . . . . . . . 10 ⊢ ;14 ∈ ℝ
21	6, 15	decnncl 12741	. . . . . . . . . . 11 ⊢ ;15 ∈ ℕ
22	21	nnrei 12265	. . . . . . . . . 10 ⊢ ;15 ∈ ℝ
23	3, 20, 22	lttri 11379	. . . . . . . . 9 ⊢ ((1 < ;14 ∧ ;14 < ;15) → 1 < ;15)
24	8, 17, 23	mp2an 690	. . . . . . . 8 ⊢ 1 < ;15
25	3, 24	ltneii 11366	. . . . . . 7 ⊢ 1 ≠ ;15
26		ccondx 17420	. . . . . . . 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 17204	. . . . 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 2754	. . . 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 2726	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
32		eqid 2726	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
33		eqid 2726	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
34		oppcbas.1	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
35	31, 32, 33, 34	oppcval 17719	. . . . 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 2785	. . 3 ⊢ (𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
38		base0 17211	. . . 4 ⊢ ∅ = (Base‘∅)
39		fvprc 6883	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
40		fvprc 6883	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅)
41	34, 40	eqtrid 2778	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅)
42	41	fveq2d 6895	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝑂) = (Base‘∅))
43	38, 39, 42	3eqtr4a 2792	. . 3 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
44	37, 43	pm2.61i 182	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
45	1, 44	eqtri 2754	1 ⊢ 𝐵 = (Base‘𝑂)

Syntax hints:  ¬ wn 3   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  Vcvv 3463  ∅c0 4323  ⟨cop 4630   class class class wbr 5144   × cxp 5671  ‘cfv 6544  (class class class)co 7414   ∈ cmpo 7416  1^st c1st 7991  2^nd c2nd 7992  tpos ctpos 8230  1c1 11148   < clt 11287  4c4 12313  5c5 12314  ;cdc 12721   sSet csts 17158  ndxcnx 17188  Basecbs 17206  Hom chom 17270  compcco 17271  oppCatcoppc 17717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-cnex 11203  ax-resscn 11204  ax-1cn 11205  ax-icn 11206  ax-addcl 11207  ax-addrcl 11208  ax-mulcl 11209  ax-mulrcl 11210  ax-mulcom 11211  ax-addass 11212  ax-mulass 11213  ax-distr 11214  ax-i2m1 11215  ax-1ne0 11216  ax-1rid 11217  ax-rnegex 11218  ax-rrecex 11219  ax-cnre 11220  ax-pre-lttri 11221  ax-pre-lttrn 11222  ax-pre-ltadd 11223  ax-pre-mulgt0 11224

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-2nd 7994  df-tpos 8231  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11289  df-mnf 11290  df-xr 11291  df-ltxr 11292  df-le 11293  df-sub 11485  df-neg 11486  df-nn 12257  df-2 12319  df-3 12320  df-4 12321  df-5 12322  df-6 12323  df-7 12324  df-8 12325  df-9 12326  df-n0 12517  df-z 12603  df-dec 12722  df-sets 17159  df-slot 17177  df-ndx 17189  df-base 17207  df-hom 17283  df-cco 17284  df-oppc 17718

This theorem is referenced by: (None)