Theorem oppcbasOLD 17764

Description: Obsolete version of oppcbas 17763 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 17247	. . . . . 6 ⊢ Base = Slot (Base‘ndx)
3		1re 11258	. . . . . . . 8 ⊢ 1 ∈ ℝ
4		1nn 12274	. . . . . . . . 9 ⊢ 1 ∈ ℕ
5		4nn0 12542	. . . . . . . . 9 ⊢ 4 ∈ ℕ0
6		1nn0 12539	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
7		1lt10 12869	. . . . . . . . 9 ⊢ 1 < ;10
8	4, 5, 6, 7	declti 12768	. . . . . . . 8 ⊢ 1 < ;14
9	3, 8	ltneii 11371	. . . . . . 7 ⊢ 1 ≠ ;14
10		basendx 17253	. . . . . . . 8 ⊢ (Base‘ndx) = 1
11		homndx 17456	. . . . . . . 8 ⊢ (Hom ‘ndx) = ;14
12	10, 11	neeq12i 3004	. . . . . . 7 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14)
13	9, 12	mpbir 231	. . . . . 6 ⊢ (Base‘ndx) ≠ (Hom ‘ndx)
14	2, 13	setsnid 17242	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩))
15		5nn 12349	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
16		4lt5 12440	. . . . . . . . . 10 ⊢ 4 < 5
17	6, 5, 15, 16	declt 12758	. . . . . . . . 9 ⊢ ;14 < ;15
18		4nn 12346	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ
19	6, 18	decnncl 12750	. . . . . . . . . . 11 ⊢ ;14 ∈ ℕ
20	19	nnrei 12272	. . . . . . . . . 10 ⊢ ;14 ∈ ℝ
21	6, 15	decnncl 12750	. . . . . . . . . . 11 ⊢ ;15 ∈ ℕ
22	21	nnrei 12272	. . . . . . . . . 10 ⊢ ;15 ∈ ℝ
23	3, 20, 22	lttri 11384	. . . . . . . . 9 ⊢ ((1 < ;14 ∧ ;14 < ;15) → 1 < ;15)
24	8, 17, 23	mp2an 692	. . . . . . . 8 ⊢ 1 < ;15
25	3, 24	ltneii 11371	. . . . . . 7 ⊢ 1 ≠ ;15
26		ccondx 17458	. . . . . . . 8 ⊢ (comp‘ndx) = ;15
27	10, 26	neeq12i 3004	. . . . . . 7 ⊢ ((Base‘ndx) ≠ (comp‘ndx) ↔ 1 ≠ ;15)
28	25, 27	mpbir 231	. . . . . 6 ⊢ (Base‘ndx) ≠ (comp‘ndx)
29	2, 28	setsnid 17242	. . . . 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 2762	. . . 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 2734	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
32		eqid 2734	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
33		eqid 2734	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
34		oppcbas.1	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
35	31, 32, 33, 34	oppcval 17757	. . . . 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 6910	. . . 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 2793	. . 3 ⊢ (𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
38		base0 17249	. . . 4 ⊢ ∅ = (Base‘∅)
39		fvprc 6898	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
40		fvprc 6898	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅)
41	34, 40	eqtrid 2786	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅)
42	41	fveq2d 6910	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝑂) = (Base‘∅))
43	38, 39, 42	3eqtr4a 2800	. . 3 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
44	37, 43	pm2.61i 182	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
45	1, 44	eqtri 2762	1 ⊢ 𝐵 = (Base‘𝑂)

Syntax hints:  ¬ wn 3   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  Vcvv 3477  ∅c0 4338  ⟨cop 4636   class class class wbr 5147   × cxp 5686  ‘cfv 6562  (class class class)co 7430   ∈ cmpo 7432  1^st c1st 8010  2^nd c2nd 8011  tpos ctpos 8248  1c1 11153   < clt 11292  4c4 12320  5c5 12321  ;cdc 12730   sSet csts 17196  ndxcnx 17226  Basecbs 17244  Hom chom 17308  compcco 17309  oppCatcoppc 17755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-hom 17321  df-cco 17322  df-oppc 17756

This theorem is referenced by: (None)