Theorem bcthlem12 8036

Description: Lemma for bcth 8058. Helper lemma for satisfying the antecendent of acdc5 7526.

Hypotheses

Ref	Expression
bcthlem12.1	⊢ D ∈ CMet
bcthlem12.3	⊢ X = dom dom D
bcthlem12.7	⊢ A = (X × {x ∈ ℝ∣0 < x})

Assertion

Ref	Expression
bcthlem12	⊢ {⟨p, r⟩∣((p ∈ X ⋀ (r ∈ ℝ ⋀ 0 < r)) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O))} ∈ ℘A

Distinct variable groups:   r,p,X   x,p,r

Proof of Theorem bcthlem12

Step	Hyp	Ref	Expression
1		opabssxp 3250	. . 3 ⊢ {⟨p, r⟩∣((p ∈ X ⋀ r ∈ {x ∈ ℝ∣0 < x}) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O))} ⊆ (X × {x ∈ ℝ∣0 < x})
2		breq2 2638	. . . . . . . 8 ⊢ (x = r → (0 < x ↔ 0 < r))
3	2	elrab 1912	. . . . . . 7 ⊢ (r ∈ {x ∈ ℝ∣0 < x} ↔ (r ∈ ℝ ⋀ 0 < r))
4	3	anbi2i 483	. . . . . 6 ⊢ ((p ∈ X ⋀ r ∈ {x ∈ ℝ∣0 < x}) ↔ (p ∈ X ⋀ (r ∈ ℝ ⋀ 0 < r)))
5	4	anbi1i 484	. . . . 5 ⊢ (((p ∈ X ⋀ r ∈ {x ∈ ℝ∣0 < x}) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O)) ↔ ((p ∈ X ⋀ (r ∈ ℝ ⋀ 0 < r)) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O)))
6	5	opabbii 2686	. . . 4 ⊢ {⟨p, r⟩∣((p ∈ X ⋀ r ∈ {x ∈ ℝ∣0 < x}) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O))} = {⟨p, r⟩∣((p ∈ X ⋀ (r ∈ ℝ ⋀ 0 < r)) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O))}
7	6	eqcomi 1486	. . 3 ⊢ {⟨p, r⟩∣((p ∈ X ⋀ (r ∈ ℝ ⋀ 0 < r)) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O))} = {⟨p, r⟩∣((p ∈ X ⋀ r ∈ {x ∈ ℝ∣0 < x}) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O))}
8		bcthlem12.7	. . 3 ⊢ A = (X × {x ∈ ℝ∣0 < x})
9	1, 7, 8	3sstr4i 2111	. 2 ⊢ {⟨p, r⟩∣((p ∈ X ⋀ (r ∈ ℝ ⋀ 0 < r)) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O))} ⊆ A
10		bcthlem12.3	. . . . . 6 ⊢ X = dom dom D
11		bcthlem12.1	. . . . . . . 8 ⊢ D ∈ CMet
12		dmexg 3374	. . . . . . . 8 ⊢ (D ∈ CMet → dom D ∈ V)
13	11, 12	ax-mp 7	. . . . . . 7 ⊢ dom D ∈ V
14	13	dmex 3376	. . . . . 6 ⊢ dom dom D ∈ V
15	10, 14	eqeltri 1551	. . . . 5 ⊢ X ∈ V
16		reex 5332	. . . . . 6 ⊢ ℝ ∈ V
17	16	rabex 2740	. . . . 5 ⊢ {x ∈ ℝ∣0 < x} ∈ V
18	15, 17	xpex 3276	. . . 4 ⊢ (X × {x ∈ ℝ∣0 < x}) ∈ V
19	8, 18	eqeltri 1551	. . 3 ⊢ A ∈ V
20	19	elpw2 2743	. 2 ⊢ ({⟨p, r⟩∣((p ∈ X ⋀ (r ∈ ℝ ⋀ 0 < r)) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O))} ∈ ℘A ↔ {⟨p, r⟩∣((p ∈ X ⋀ (r ∈ ℝ ⋀ 0 < r)) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O))} ⊆ A)
21	9, 20	mpbir 190	1 ⊢ {⟨p, r⟩∣((p ∈ X ⋀ (r ∈ ℝ ⋀ 0 < r)) ⋀ (r < ((2nd ‘y) / 2) ⋀ (p( ball ‘D)r) ⊆ O))} ∈ ℘A

Syntax hints:   ⋀ wa 223   = wceq 960   ∈ wcel 962  {crab 1655  Vcvv 1818   ⊆ wss 2058  ℘cpw 2413   class class class wbr 2634  {copab 2681   × cxp 3184  dom cdm 3186   ‘cfv 3198  (class class class)co 3979  2^nd c2nd 4094  ℝcr 5253  0cc0 5254   / cdiv 5314   < clt 5506  2c2 5975   ball cbl 7817  CMetcms 7947

This theorem is referenced by:  bcthlem30 8054

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-rep 2708  ax-sep 2718  ax-nul 2725  ax-pow 2758  ax-pr 2795  ax-un 2882  ax-inf2 4642

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 780  df-3an 781  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-rex 1657  df-reu 1658  df-rab 1659  df-v 1819  df-sbc 1949  df-csb 2012  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-pss 2066  df-nul 2292  df-if 2374  df-pw 2414  df-sn 2424  df-pr 2425  df-tp 2427  df-op 2428  df-uni 2518  df-int 2548  df-iun 2582  df-br 2635  df-opab 2682  df-tr 2696  df-eprel 2848  df-id 2851  df-po 2856  df-so 2866  df-fr 2933  df-we 2950  df-ord 2967  df-on 2968  df-lim 2969  df-suc 2970  df-om 3148  df-xp 3200  df-rel 3201  df-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fn 3209  df-f 3210  df-fv 3214  df-rdg 3948  df-opr 3981  df-oprab 3982  df-1st 4095  df-2nd 4096  df-1o 4149  df-oadd 4151  df-omul 4152  df-er 4277  df-ec 4279  df-qs 4282  df-ni 5020  df-pli 5021  df-mi 5022  df-lti 5023  df-plpq 5055  df-mpq 5056  df-enq 5057  df-nq 5058  df-plq 5059  df-mq 5060  df-rq 5061  df-ltq 5062  df-1q 5063  df-np 5106  df-1p 5107  df-enr 5186  df-nr 5187  df-0r 5191  df-c 5260  df-r 5264

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