Theorem oppcbas 16201

Description: Base set of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
oppcbas	⊢ 𝐵 = (Base‘𝑂)

Proof of Theorem oppcbas

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		eqid 2610	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
3		eqid 2610	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2610	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
5		oppcbas.1	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
6	2, 3, 4, 5	oppcval 16196	. . . . 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)))⟩))
7	6	fveq2d 6107	. . . 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)))⟩)))
8		baseid 15747	. . . . . 6 ⊢ Base = Slot (Base‘ndx)
9		1re 9918	. . . . . . . 8 ⊢ 1 ∈ ℝ
10		1nn 10908	. . . . . . . . 9 ⊢ 1 ∈ ℕ
11		4nn0 11188	. . . . . . . . 9 ⊢ 4 ∈ ℕ0
12		1nn0 11185	. . . . . . . . 9 ⊢ 1 ∈ ℕ0
13		1lt10 11557	. . . . . . . . 9 ⊢ 1 < ;10
14	10, 11, 12, 13	declti 11422	. . . . . . . 8 ⊢ 1 < ;14
15	9, 14	ltneii 10029	. . . . . . 7 ⊢ 1 ≠ ;14
16		basendx 15751	. . . . . . . 8 ⊢ (Base‘ndx) = 1
17		homndx 15897	. . . . . . . 8 ⊢ (Hom ‘ndx) = ;14
18	16, 17	neeq12i 2848	. . . . . . 7 ⊢ ((Base‘ndx) ≠ (Hom ‘ndx) ↔ 1 ≠ ;14)
19	15, 18	mpbir 220	. . . . . 6 ⊢ (Base‘ndx) ≠ (Hom ‘ndx)
20	8, 19	setsnid 15743	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(𝐶 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝐶)⟩))
21		5nn 11065	. . . . . . . . . 10 ⊢ 5 ∈ ℕ
22		4lt5 11077	. . . . . . . . . 10 ⊢ 4 < 5
23	12, 11, 21, 22	declt 11406	. . . . . . . . 9 ⊢ ;14 < ;15
24		4nn 11064	. . . . . . . . . . . 12 ⊢ 4 ∈ ℕ
25	12, 24	decnncl 11394	. . . . . . . . . . 11 ⊢ ;14 ∈ ℕ
26	25	nnrei 10906	. . . . . . . . . 10 ⊢ ;14 ∈ ℝ
27	12, 21	decnncl 11394	. . . . . . . . . . 11 ⊢ ;15 ∈ ℕ
28	27	nnrei 10906	. . . . . . . . . 10 ⊢ ;15 ∈ ℝ
29	9, 26, 28	lttri 10042	. . . . . . . . 9 ⊢ ((1 < ;14 ∧ ;14 < ;15) → 1 < ;15)
30	14, 23, 29	mp2an 704	. . . . . . . 8 ⊢ 1 < ;15
31	9, 30	ltneii 10029	. . . . . . 7 ⊢ 1 ≠ ;15
32		ccondx 15899	. . . . . . . 8 ⊢ (comp‘ndx) = ;15
33	16, 32	neeq12i 2848	. . . . . . 7 ⊢ ((Base‘ndx) ≠ (comp‘ndx) ↔ 1 ≠ ;15)
34	31, 33	mpbir 220	. . . . . 6 ⊢ (Base‘ndx) ≠ (comp‘ndx)
35	8, 34	setsnid 15743	. . . . 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)))⟩))
36	20, 35	eqtri 2632	. . . 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)))⟩))
37	7, 36	syl6reqr 2663	. . 3 ⊢ (𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
38		base0 15740	. . . 4 ⊢ ∅ = (Base‘∅)
39		fvprc 6097	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
40		fvprc 6097	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅)
41	5, 40	syl5eq 2656	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅)
42	41	fveq2d 6107	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Base‘𝑂) = (Base‘∅))
43	38, 39, 42	3eqtr4a 2670	. . 3 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = (Base‘𝑂))
44	37, 43	pm2.61i 175	. 2 ⊢ (Base‘𝐶) = (Base‘𝑂)
45	1, 44	eqtri 2632	1 ⊢ 𝐵 = (Base‘𝑂)

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058  tpos ctpos 7238  1c1 9816   < clt 9953  4c4 10949  5c5 10950  ;cdc 11369  ndxcnx 15692   sSet csts 15693  Basecbs 15695  Hom chom 15779  compcco 15780  oppCatcoppc 16194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-hom 15793  df-cco 15794  df-oppc 16195

This theorem is referenced by:  oppccatid  16202  oppchomf  16203  2oppcbas  16206  2oppccomf  16208  oppccomfpropd  16210  isepi  16223  epii  16226  oppcsect  16261  oppcsect2  16262  oppcinv  16263  oppciso  16264  sectepi  16267  episect  16268  funcoppc  16358  fulloppc  16405  fthoppc  16406  fthepi  16411  hofcl  16722  yon11  16727  yon12  16728  yon2  16729  oyon1cl  16734  yonedalem21  16736  yonedalem3a  16737  yonedalem4c  16740  yonedalem22  16741  yonedalem3b  16742  yonedalem3  16743  yonedainv  16744  yonffthlem  16745