Theorem oppcbasOLD 17321

Description: Obsolete version of oppcbas 17320 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 16818	. . . . . 6 ⊢ Base = Slot (Base‘ndx)
3		1re 10881	. . . . . . . 8 ⊢ 1 ∈ ℝ
4		1nn 11889	. . . . . . . . 9 ⊢ 1 ∈ ℕ
5		4nn0 12157	. . . . . . . . 9 ⊢ 4 ∈ ℕ0
6		1nn0 12154	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
7		1lt10 12480	. . . . . . . . 9 ⊢ 1 < ;10
8	4, 5, 6, 7	declti 12379	. . . . . . . 8 ⊢ 1 < ;14
9	3, 8	ltneii 10993	. . . . . . 7 ⊢ 1 ≠ ;14
10		basendx 16824	. . . . . . . 8 ⊢ (Base‘ndx) = 1
11		homndx 17015	. . . . . . . 8 ⊢ (Hom ‘ndx) = ;14
12	10, 11	neeq12i 3010	. . . . . . 7 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14)
13	9, 12	mpbir 234	. . . . . 6 ⊢ (Base‘ndx) ≠ (Hom ‘ndx)
14	2, 13	setsnid 16813	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩))
15		5nn 11964	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
16		4lt5 12055	. . . . . . . . . 10 ⊢ 4 < 5
17	6, 5, 15, 16	declt 12369	. . . . . . . . 9 ⊢ ;14 < ;15
18		4nn 11961	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ
19	6, 18	decnncl 12361	. . . . . . . . . . 11 ⊢ ;14 ∈ ℕ
20	19	nnrei 11887	. . . . . . . . . 10 ⊢ ;14 ∈ ℝ
21	6, 15	decnncl 12361	. . . . . . . . . . 11 ⊢ ;15 ∈ ℕ
22	21	nnrei 11887	. . . . . . . . . 10 ⊢ ;15 ∈ ℝ
23	3, 20, 22	lttri 11006	. . . . . . . . 9 ⊢ ((1 < ;14 ∧ ;14 < ;15) → 1 < ;15)
24	8, 17, 23	mp2an 692	. . . . . . . 8 ⊢ 1 < ;15
25	3, 24	ltneii 10993	. . . . . . 7 ⊢ 1 ≠ ;15
26		ccondx 17017	. . . . . . . 8 ⊢ (comp‘ndx) = ;15
27	10, 26	neeq12i 3010	. . . . . . 7 ⊢ ((Base‘ndx) ≠ (comp‘ndx) ↔ 1 ≠ ;15)
28	25, 27	mpbir 234	. . . . . 6 ⊢ (Base‘ndx) ≠ (comp‘ndx)
29	2, 28	setsnid 16813	. . . . 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 2767	. . . 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 2739	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
32		eqid 2739	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
33		eqid 2739	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
34		oppcbas.1	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
35	31, 32, 33, 34	oppcval 17314	. . . . 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 6757	. . . 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 2799	. . 3 ⊢ (𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
38		base0 16820	. . . 4 ⊢ ∅ = (Base‘∅)
39		fvprc 6745	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
40		fvprc 6745	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅)
41	34, 40	eqtrid 2791	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅)
42	41	fveq2d 6757	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝑂) = (Base‘∅))
43	38, 39, 42	3eqtr4a 2806	. . 3 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
44	37, 43	pm2.61i 185	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
45	1, 44	eqtri 2767	1 ⊢ 𝐵 = (Base‘𝑂)

Syntax hints:  ¬ wn 3   = wceq 1543   ∈ wcel 2112   ≠ wne 2943  Vcvv 3423  ∅c0 4254  ⟨cop 4564   class class class wbr 5070   × cxp 5577  ‘cfv 6415  (class class class)co 7252   ∈ cmpo 7254  1^st c1st 7799  2^nd c2nd 7800  tpos ctpos 8009  1c1 10778   < clt 10915  4c4 11935  5c5 11936  ;cdc 12341   sSet csts 16767  ndxcnx 16797  Basecbs 16815  Hom chom 16874  compcco 16875  oppCatcoppc 17312

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 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5216  ax-nul 5223  ax-pow 5282  ax-pr 5346  ax-un 7563  ax-cnex 10833  ax-resscn 10834  ax-1cn 10835  ax-icn 10836  ax-addcl 10837  ax-addrcl 10838  ax-mulcl 10839  ax-mulrcl 10840  ax-mulcom 10841  ax-addass 10842  ax-mulass 10843  ax-distr 10844  ax-i2m1 10845  ax-1ne0 10846  ax-1rid 10847  ax-rnegex 10848  ax-rrecex 10849  ax-cnre 10850  ax-pre-lttri 10851  ax-pre-lttrn 10852  ax-pre-ltadd 10853  ax-pre-mulgt0 10854

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 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3425  df-sbc 3713  df-csb 3830  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4255  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 5153  df-tr 5186  df-id 5479  df-eprel 5485  df-po 5493  df-so 5494  df-fr 5534  df-we 5536  df-xp 5585  df-rel 5586  df-cnv 5587  df-co 5588  df-dm 5589  df-rn 5590  df-res 5591  df-ima 5592  df-pred 6189  df-ord 6251  df-on 6252  df-lim 6253  df-suc 6254  df-iota 6373  df-fun 6417  df-fn 6418  df-f 6419  df-f1 6420  df-fo 6421  df-f1o 6422  df-fv 6423  df-riota 7209  df-ov 7255  df-oprab 7256  df-mpo 7257  df-om 7685  df-tpos 8010  df-wrecs 8089  df-recs 8150  df-rdg 8188  df-er 8433  df-en 8669  df-dom 8670  df-sdom 8671  df-pnf 10917  df-mnf 10918  df-xr 10919  df-ltxr 10920  df-le 10921  df-sub 11112  df-neg 11113  df-nn 11879  df-2 11941  df-3 11942  df-4 11943  df-5 11944  df-6 11945  df-7 11946  df-8 11947  df-9 11948  df-n0 12139  df-z 12225  df-dec 12342  df-sets 16768  df-slot 16786  df-ndx 16798  df-base 16816  df-hom 16887  df-cco 16888  df-oppc 17313

This theorem is referenced by: (None)