Theorem cvrfval 39261

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 3468	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		cvrfval.c	. . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾)
3		fveq2 6858	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4		cvrfval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6	5	eleq2d 2814	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥 ∈ 𝐵))
7	5	eleq2d 2814	. . . . . . 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 6858	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (lt‘𝑝) = (lt‘𝐾))
10		cvrfval.s	. . . . . . . 8 ⊢ < = (lt‘𝐾)
11	9, 10	eqtr4di 2782	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (lt‘𝑝) = < )
12	11	breqd 5118	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑦 ↔ 𝑥 < 𝑦))
13	11	breqd 5118	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑧 ↔ 𝑥 < 𝑧))
14	11	breqd 5118	. . . . . . . . 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 3320	. . . . . . 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 5173	. . . 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 39259	. . . 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 5174	. . . . 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 6872	. . . . . . 7 ⊢ 𝐵 ∈ V
24	23, 23	xpex 7729	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
25		opabssxp 5731	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))} ⊆ (𝐵 × 𝐵)
26	24, 25	ssexi 5277	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))} ∈ V
27	22, 26	eqeltri 2824	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))} ∈ V
28	19, 20, 27	fvmpt 6968	. . 3 ⊢ (𝐾 ∈ V → ( ⋖ ‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))})
29	2, 28	eqtrid 2776	. 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 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447   class class class wbr 5107  {copab 5169   × cxp 5636  ‘cfv 6511  Basecbs 17179  ltcplt 18269   ⋖ ccvr 39255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-covers 39259

This theorem is referenced by:  cvrval  39262