Theorem cvrfval 40517

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 3475	. 2 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
2		cvrfval.c	. . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾)
3		fveq2 6908	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4		cvrfval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
5	3, 4	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6	5	eleq2d 2850	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (𝑥 ∈ (Base‘𝑝) ↔ 𝑥 ∈ 𝐵))
7	5	eleq2d 2850	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (𝑦 ∈ (Base‘𝑝) ↔ 𝑦 ∈ 𝐵))
8	6, 7	anbi12d 644	. . . . . 6 ⊢ (𝑝 = 𝐾 → ((𝑥 ∈ (Base‘𝑝) ∧ 𝑦 ∈ (Base‘𝑝)) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
9		fveq2 6908	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → (lt‘𝑝) = (lt‘𝐾))
10		cvrfval.s	. . . . . . . 8 ⊢ < = (lt‘𝐾)
11	9, 10	eqtr4di 2816	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (lt‘𝑝) = < )
12	11	breqd 5117	. . . . . 6 ⊢ (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑦 ↔ 𝑥 < 𝑦))
13	11	breqd 5117	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (𝑥(lt‘𝑝)𝑧 ↔ 𝑥 < 𝑧))
14	11	breqd 5117	. . . . . . . . 9 ⊢ (𝑝 = 𝐾 → (𝑧(lt‘𝑝)𝑦 ↔ 𝑧 < 𝑦))
15	13, 14	anbi12d 644	. . . . . . . 8 ⊢ (𝑝 = 𝐾 → ((𝑥(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑦) ↔ (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))
16	5, 15	rexeqbidv 3339	. . . . . . 7 ⊢ (𝑝 = 𝐾 → (∃𝑧 ∈ (Base‘𝑝)(𝑥(lt‘𝑝)𝑧 ∧ 𝑧(lt‘𝑝)𝑦) ↔ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))
17	16	notbid 321	. . . . . 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 1465	. . . . 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 5174	. . . 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 40515	. . . 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 1111	. . . . . 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 5175	. . . . 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 6922	. . . . . . 7 ⊢ 𝐵 ∈ V
24	23, 23	xpex 7791	. . . . . 6 ⊢ (𝐵 × 𝐵) ∈ V
25		opabssxp 5748	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))} ⊆ (𝐵 × 𝐵)
26	24, 25	ssexi 5288	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦)))} ∈ V
27	22, 26	eqeltri 2860	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))} ∈ V
28	19, 20, 27	fvmpt 7016	. . 3 ⊢ (𝐾 ∈ V → ( ⋖ ‘𝐾) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))})
29	2, 28	eqtrid 2810	. 2 ⊢ (𝐾 ∈ V → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))})
30	1, 29	syl 18	1 ⊢ (𝐾 ∈ 𝐴 → 𝐶 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 < 𝑦 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑥 < 𝑧 ∧ 𝑧 < 𝑦))})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  Vcvv 3454   class class class wbr 5106  {copab 5170   × cxp 5653  ‘cfv 6558  Basecbs 17416  ltcplt 18511   ⋖ ccvr 40511

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-covers 40515

This theorem is used by:  cvrval  40518