Theorem oppcbasOLD 17664

Description: Obsolete version of oppcbas 17663 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 17147	. . . . . 6 ⊢ Base = Slot (Base‘ndx)
3		1re 11214	. . . . . . . 8 ⊢ 1 ∈ ℝ
4		1nn 12223	. . . . . . . . 9 ⊢ 1 ∈ ℕ
5		4nn0 12491	. . . . . . . . 9 ⊢ 4 ∈ ℕ0
6		1nn0 12488	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
7		1lt10 12816	. . . . . . . . 9 ⊢ 1 < ;10
8	4, 5, 6, 7	declti 12715	. . . . . . . 8 ⊢ 1 < ;14
9	3, 8	ltneii 11327	. . . . . . 7 ⊢ 1 ≠ ;14
10		basendx 17153	. . . . . . . 8 ⊢ (Base‘ndx) = 1
11		homndx 17356	. . . . . . . 8 ⊢ (Hom ‘ndx) = ;14
12	10, 11	neeq12i 3008	. . . . . . 7 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14)
13	9, 12	mpbir 230	. . . . . 6 ⊢ (Base‘ndx) ≠ (Hom ‘ndx)
14	2, 13	setsnid 17142	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩))
15		5nn 12298	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
16		4lt5 12389	. . . . . . . . . 10 ⊢ 4 < 5
17	6, 5, 15, 16	declt 12705	. . . . . . . . 9 ⊢ ;14 < ;15
18		4nn 12295	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ
19	6, 18	decnncl 12697	. . . . . . . . . . 11 ⊢ ;14 ∈ ℕ
20	19	nnrei 12221	. . . . . . . . . 10 ⊢ ;14 ∈ ℝ
21	6, 15	decnncl 12697	. . . . . . . . . . 11 ⊢ ;15 ∈ ℕ
22	21	nnrei 12221	. . . . . . . . . 10 ⊢ ;15 ∈ ℝ
23	3, 20, 22	lttri 11340	. . . . . . . . 9 ⊢ ((1 < ;14 ∧ ;14 < ;15) → 1 < ;15)
24	8, 17, 23	mp2an 691	. . . . . . . 8 ⊢ 1 < ;15
25	3, 24	ltneii 11327	. . . . . . 7 ⊢ 1 ≠ ;15
26		ccondx 17358	. . . . . . . 8 ⊢ (comp‘ndx) = ;15
27	10, 26	neeq12i 3008	. . . . . . 7 ⊢ ((Base‘ndx) ≠ (comp‘ndx) ↔ 1 ≠ ;15)
28	25, 27	mpbir 230	. . . . . 6 ⊢ (Base‘ndx) ≠ (comp‘ndx)
29	2, 28	setsnid 17142	. . . . 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 2761	. . . 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 2733	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
32		eqid 2733	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
33		eqid 2733	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
34		oppcbas.1	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
35	31, 32, 33, 34	oppcval 17657	. . . . 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 6896	. . . 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 2792	. . 3 ⊢ (𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
38		base0 17149	. . . 4 ⊢ ∅ = (Base‘∅)
39		fvprc 6884	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
40		fvprc 6884	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅)
41	34, 40	eqtrid 2785	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅)
42	41	fveq2d 6896	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝑂) = (Base‘∅))
43	38, 39, 42	3eqtr4a 2799	. . 3 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
44	37, 43	pm2.61i 182	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
45	1, 44	eqtri 2761	1 ⊢ 𝐵 = (Base‘𝑂)

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  Vcvv 3475  ∅c0 4323  ⟨cop 4635   class class class wbr 5149   × cxp 5675  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1^st c1st 7973  2^nd c2nd 7974  tpos ctpos 8210  1c1 11111   < clt 11248  4c4 12269  5c5 12270  ;cdc 12677   sSet csts 17096  ndxcnx 17126  Basecbs 17144  Hom chom 17208  compcco 17209  oppCatcoppc 17655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  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 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-hom 17221  df-cco 17222  df-oppc 17656

This theorem is referenced by: (None)