Theorem cvrfval 39306

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 3457	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		cvrfval.c	. . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾)
3		fveq2 6822	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4		cvrfval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2784	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6	5	eleq2d 2817	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥 ∈ 𝐵))
7	5	eleq2d 2817	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦 ∈ 𝐵))
8	6, 7	anbi12d 632	. . . . . 6 ⊢ (𝑝 = 𝐾 → ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
9		fveq2 6822	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (lt‘𝑝) = (lt‘𝐾))
10		cvrfval.s	. . . . . . . 8 ⊢ < = (lt‘𝐾)
11	9, 10	eqtr4di 2784	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (lt‘𝑝) = < )
12	11	breqd 5102	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑦 ↔ 𝑥 < 𝑦))
13	11	breqd 5102	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑧 ↔ 𝑥 < 𝑧))
14	11	breqd 5102	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (𝑧(lt‘𝑝)𝑦 ↔ 𝑧 < 𝑦))
15	13, 14	anbi12d 632	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → ((𝑥(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑦) ↔ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))
16	5, 15	rexeqbidv 3313	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑦) ↔ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))
17	16	notbid 318	. . . . . 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 1438	. . . . 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 5157	. . . 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 39304	. . . 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 1094	. . . . . 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 5158	. . . . 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 6836	. . . . . . 7 ⊢ 𝐵 ∈ V
24	23, 23	xpex 7686	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
25		opabssxp 5708	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))} ⊆ (𝐵 × 𝐵)
26	24, 25	ssexi 5260	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))} ∈ V
27	22, 26	eqeltri 2827	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))} ∈ V
28	19, 20, 27	fvmpt 6929	. . 3 ⊢ (𝐾 ∈ V → ( ⋖ ‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))})
29	2, 28	eqtrid 2778	. 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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  Vcvv 3436   class class class wbr 5091  {copab 5153   × cxp 5614  ‘cfv 6481  Basecbs 17117  ltcplt 18211   ⋖ ccvr 39300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-covers 39304

This theorem is referenced by:  cvrval  39307