Theorem cvrfval 35878

Description: Value of covers relation "is covered by". (Contributed by NM, 18-Sep-2011.)

Hypotheses

Ref	Expression
cvrfval.b	⊢ 𝐵 = (Base‘𝐾)
cvrfval.s	⊢ < = (lt‘𝐾)
cvrfval.c	⊢ 𝐶 = ( ⋖ ‘𝐾)

Assertion

Ref	Expression
cvrfval	⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))})

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐾,𝑦,𝑧

Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)   < (𝑥,𝑦,𝑧)

Proof of Theorem cvrfval

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3427	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		cvrfval.c	. . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾)
3		fveq2 6496	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4		cvrfval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	syl6eqr 2826	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6	5	eleq2d 2845	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥 ∈ 𝐵))
7	5	eleq2d 2845	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦 ∈ 𝐵))
8	6, 7	anbi12d 621	. . . . . 6 ⊢ (𝑝 = 𝐾 → ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
9		fveq2 6496	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (lt‘𝑝) = (lt‘𝐾))
10		cvrfval.s	. . . . . . . 8 ⊢ < = (lt‘𝐾)
11	9, 10	syl6eqr 2826	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (lt‘𝑝) = < )
12	11	breqd 4936	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑦 ↔ 𝑥 < 𝑦))
13	11	breqd 4936	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑧 ↔ 𝑥 < 𝑧))
14	11	breqd 4936	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (𝑧(lt‘𝑝)𝑦 ↔ 𝑧 < 𝑦))
15	13, 14	anbi12d 621	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → ((𝑥(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑦) ↔ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))
16	5, 15	rexeqbidv 3336	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑦) ↔ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))
17	16	notbid 310	. . . . . 6 ⊢ (𝑝 = 𝐾 → (¬ ∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑦) ↔ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))
18	8, 12, 17	3anbi123d 1415	. . . . 5 ⊢ (𝑝 = 𝐾 → (((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ∧ 𝑥(lt‘𝑝)𝑦 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑦)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))))
19	18	opabbidv 4991	. . . 4 ⊢ (𝑝 = 𝐾 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ∧ 𝑥(lt‘𝑝)𝑦 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))})
20		df-covers 35876	. . . 4 ⊢ ⋖ = (𝑝 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ∧ 𝑥(lt‘𝑝)𝑦 ∧ ¬ ∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑦))})
21		3anass 1076	. . . . . 6 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))))
22	21	opabbii 4992	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))}
23	4	fvexi 6510	. . . . . . 7 ⊢ 𝐵 ∈ V
24	23, 23	xpex 7291	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
25		opabssxp 5489	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))} ⊆ (𝐵 × 𝐵)
26	24, 25	ssexi 5078	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))} ∈ V
27	22, 26	eqeltri 2856	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))} ∈ V
28	19, 20, 27	fvmpt 6593	. . 3 ⊢ (𝐾 ∈ V → ( ⋖ ‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))})
29	2, 28	syl5eq 2820	. 2 ⊢ (𝐾 ∈ V → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))})
30	1, 29	syl 17	1 ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387   ∧ w3a 1068   = wceq 1507   ∈ wcel 2050  ∃wrex 3083  Vcvv 3409   class class class wbr 4925  {copab 4987   × cxp 5401  ‘cfv 6185  Basecbs 16337  ltcplt 17421   ⋖ ccvr 35872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193  df-covers 35876

This theorem is referenced by:  cvrval  35879