Theorem oppchomfval 16197

Description: Hom-sets of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
oppchom.h	⊢ 𝐻 = (Hom ‘𝐶)
oppchom.o	⊢ 𝑂 = (oppCat‘𝐶)

Assertion

Ref	Expression
oppchomfval	⊢ tpos 𝐻 = (Hom ‘𝑂)

Proof of Theorem oppchomfval

Dummy variables 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		homid 15898	. . . 4 ⊢ Hom = Slot (Hom ‘ndx)
2		1nn0 11185	. . . . . . . 8 ⊢ 1 ∈ ℕ0
3		4nn 11064	. . . . . . . 8 ⊢ 4 ∈ ℕ
4	2, 3	decnncl 11394	. . . . . . 7 ⊢ ;14 ∈ ℕ
5	4	nnrei 10906	. . . . . 6 ⊢ ;14 ∈ ℝ
6		4nn0 11188	. . . . . . 7 ⊢ 4 ∈ ℕ0
7		5nn 11065	. . . . . . 7 ⊢ 5 ∈ ℕ
8		4lt5 11077	. . . . . . 7 ⊢ 4 < 5
9	2, 6, 7, 8	declt 11406	. . . . . 6 ⊢ ;14 < ;15
10	5, 9	ltneii 10029	. . . . 5 ⊢ ;14 ≠ ;15
11		homndx 15897	. . . . . 6 ⊢ (Hom ‘ndx) = ;14
12		ccondx 15899	. . . . . 6 ⊢ (comp‘ndx) = ;15
13	11, 12	neeq12i 2848	. . . . 5 ⊢ ((Hom ‘ndx) ≠ (comp‘ndx) ↔ ;14 ≠ ;15)
14	10, 13	mpbir 220	. . . 4 ⊢ (Hom ‘ndx) ≠ (comp‘ndx)
15	1, 14	setsnid 15743	. . 3 ⊢ (Hom ‘(𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩)) = (Hom ‘((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑢)))⟩))
16		oppchom.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
17		fvex 6113	. . . . . 6 ⊢ (Hom ‘𝐶) ∈ V
18	16, 17	eqeltri 2684	. . . . 5 ⊢ 𝐻 ∈ V
19	18	tposex 7273	. . . 4 ⊢ tpos 𝐻 ∈ V
20	1	setsid 15742	. . . 4 ⊢ ((𝐶 ∈ V ∧ tpos 𝐻 ∈ V) → tpos 𝐻 = (Hom ‘(𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩)))
21	19, 20	mpan2 703	. . 3 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘(𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩)))
22		eqid 2610	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
23		eqid 2610	. . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶)
24		oppchom.o	. . . . 5 ⊢ 𝑂 = (oppCat‘𝐶)
25	22, 16, 23, 24	oppcval 16196	. . . 4 ⊢ (𝐶 ∈ V → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑢)))⟩))
26	25	fveq2d 6107	. . 3 ⊢ (𝐶 ∈ V → (Hom ‘𝑂) = (Hom ‘((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑧 ∈ (Base‘𝐶) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝐶)(1st ‘𝑢)))⟩)))
27	15, 21, 26	3eqtr4a 2670	. 2 ⊢ (𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂))
28		tpos0 7269	. . 3 ⊢ tpos ∅ = ∅
29		fvprc 6097	. . . . 5 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝐶) = ∅)
30	16, 29	syl5eq 2656	. . . 4 ⊢ (¬ 𝐶 ∈ V → 𝐻 = ∅)
31	30	tposeqd 7242	. . 3 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = tpos ∅)
32		fvprc 6097	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → (oppCat‘𝐶) = ∅)
33	24, 32	syl5eq 2656	. . . . 5 ⊢ (¬ 𝐶 ∈ V → 𝑂 = ∅)
34	33	fveq2d 6107	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = (Hom ‘∅))
35		df-hom 15793	. . . . 5 ⊢ Hom = Slot ;14
36	35	str0 15739	. . . 4 ⊢ ∅ = (Hom ‘∅)
37	34, 36	syl6eqr 2662	. . 3 ⊢ (¬ 𝐶 ∈ V → (Hom ‘𝑂) = ∅)
38	28, 31, 37	3eqtr4a 2670	. 2 ⊢ (¬ 𝐶 ∈ V → tpos 𝐻 = (Hom ‘𝑂))
39	27, 38	pm2.61i 175	1 ⊢ tpos 𝐻 = (Hom ‘𝑂)

Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ∅c0 3874  ⟨cop 4131   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058  tpos ctpos 7238  1c1 9816  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-dec 11370  df-ndx 15698  df-slot 15699  df-sets 15701  df-hom 15793  df-cco 15794  df-oppc 16195

This theorem is referenced by:  oppchom  16198