Theorem alexsubALTlem1 24003

Description: Lemma for alexsubALT 24007. A compact space has a subbase such that every cover taken from it has a finite subcover. (Contributed by Jeff Hankins, 27-Jan-2010.)

Hypothesis

Ref	Expression
alexsubALT.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
alexsubALTlem1	⊢ (𝐽 ∈ Comp → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))

Distinct variable groups:   𝑐,𝑑,𝑥,𝐽   𝑋,𝑐,𝑑,𝑥

Proof of Theorem alexsubALTlem1

Step	Hyp	Ref	Expression
1		cmptop 23351	. . 3 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
2		fitop 22856	. . . . 5 ⊢ (𝐽 ∈ Top → (fi‘𝐽) = 𝐽)
3	2	fveq2d 6846	. . . 4 ⊢ (𝐽 ∈ Top → (topGen‘(fi‘𝐽)) = (topGen‘𝐽))
4		tgtop 22929	. . . 4 ⊢ (𝐽 ∈ Top → (topGen‘𝐽) = 𝐽)
5	3, 4	eqtr2d 2773	. . 3 ⊢ (𝐽 ∈ Top → 𝐽 = (topGen‘(fi‘𝐽)))
6	1, 5	syl 17	. 2 ⊢ (𝐽 ∈ Comp → 𝐽 = (topGen‘(fi‘𝐽)))
7		velpw 4561	. . . 4 ⊢ (𝑐 ∈ 𝒫 𝐽 ↔ 𝑐 ⊆ 𝐽)
8		alexsubALT.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
9	8	cmpcov 23345	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)
10	9	3exp 1120	. . . 4 ⊢ (𝐽 ∈ Comp → (𝑐 ⊆ 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
11	7, 10	biimtrid 242	. . 3 ⊢ (𝐽 ∈ Comp → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
12	11	ralrimiv 3129	. 2 ⊢ (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))
13		2fveq3 6847	. . . . 5 ⊢ (𝑥 = 𝐽 → (topGen‘(fi‘𝑥)) = (topGen‘(fi‘𝐽)))
14	13	eqeq2d 2748	. . . 4 ⊢ (𝑥 = 𝐽 → (𝐽 = (topGen‘(fi‘𝑥)) ↔ 𝐽 = (topGen‘(fi‘𝐽))))
15		pweq 4570	. . . . 5 ⊢ (𝑥 = 𝐽 → 𝒫 𝑥 = 𝒫 𝐽)
16	15	raleqdv 3298	. . . 4 ⊢ (𝑥 = 𝐽 → (∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑) ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))
17	14, 16	anbi12d 633	. . 3 ⊢ (𝑥 = 𝐽 → ((𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)) ↔ (𝐽 = (topGen‘(fi‘𝐽)) ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))))
18	17	spcegv 3553	. 2 ⊢ (𝐽 ∈ Comp → ((𝐽 = (topGen‘(fi‘𝐽)) ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)) → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑))))
19	6, 12, 18	mp2and 700	1 ⊢ (𝐽 ∈ Comp → ∃𝑥(𝐽 = (topGen‘(fi‘𝑥)) ∧ ∀𝑐 ∈ 𝒫 𝑥(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3902   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865  ‘cfv 6500  Fincfn 8895  ficfi 9325  topGenctg 17369  Topctop 22849  Compccmp 23342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-1o 8407  df-2o 8408  df-en 8896  df-fin 8899  df-fi 9326  df-topgen 17375  df-top 22850  df-cmp 23343

This theorem is referenced by:  alexsubALT  24007