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Theorem alexsubALTlem1 23421
Description: Lemma for alexsubALT 23425. 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 = βˆͺ 𝐽
Assertion
Ref Expression
alexsubALTlem1 (𝐽 ∈ Comp β†’ βˆƒπ‘₯(𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑)))
Distinct variable groups:   𝑐,𝑑,π‘₯,𝐽   𝑋,𝑐,𝑑,π‘₯

Proof of Theorem alexsubALTlem1
StepHypRef Expression
1 cmptop 22769 . . 3 (𝐽 ∈ Comp β†’ 𝐽 ∈ Top)
2 fitop 22272 . . . . 5 (𝐽 ∈ Top β†’ (fiβ€˜π½) = 𝐽)
32fveq2d 6850 . . . 4 (𝐽 ∈ Top β†’ (topGenβ€˜(fiβ€˜π½)) = (topGenβ€˜π½))
4 tgtop 22346 . . . 4 (𝐽 ∈ Top β†’ (topGenβ€˜π½) = 𝐽)
53, 4eqtr2d 2774 . . 3 (𝐽 ∈ Top β†’ 𝐽 = (topGenβ€˜(fiβ€˜π½)))
61, 5syl 17 . 2 (𝐽 ∈ Comp β†’ 𝐽 = (topGenβ€˜(fiβ€˜π½)))
7 velpw 4569 . . . 4 (𝑐 ∈ 𝒫 𝐽 ↔ 𝑐 βŠ† 𝐽)
8 alexsubALT.1 . . . . . 6 𝑋 = βˆͺ 𝐽
98cmpcov 22763 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑐 βŠ† 𝐽 ∧ 𝑋 = βˆͺ 𝑐) β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑)
1093exp 1120 . . . 4 (𝐽 ∈ Comp β†’ (𝑐 βŠ† 𝐽 β†’ (𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑)))
117, 10biimtrid 241 . . 3 (𝐽 ∈ Comp β†’ (𝑐 ∈ 𝒫 𝐽 β†’ (𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑)))
1211ralrimiv 3139 . 2 (𝐽 ∈ Comp β†’ βˆ€π‘ ∈ 𝒫 𝐽(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑))
13 2fveq3 6851 . . . . 5 (π‘₯ = 𝐽 β†’ (topGenβ€˜(fiβ€˜π‘₯)) = (topGenβ€˜(fiβ€˜π½)))
1413eqeq2d 2744 . . . 4 (π‘₯ = 𝐽 β†’ (𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ↔ 𝐽 = (topGenβ€˜(fiβ€˜π½))))
15 pweq 4578 . . . . 5 (π‘₯ = 𝐽 β†’ 𝒫 π‘₯ = 𝒫 𝐽)
1615raleqdv 3312 . . . 4 (π‘₯ = 𝐽 β†’ (βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ↔ βˆ€π‘ ∈ 𝒫 𝐽(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑)))
1714, 16anbi12d 632 . . 3 (π‘₯ = 𝐽 β†’ ((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑)) ↔ (𝐽 = (topGenβ€˜(fiβ€˜π½)) ∧ βˆ€π‘ ∈ 𝒫 𝐽(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑))))
1817spcegv 3558 . 2 (𝐽 ∈ Comp β†’ ((𝐽 = (topGenβ€˜(fiβ€˜π½)) ∧ βˆ€π‘ ∈ 𝒫 𝐽(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑)) β†’ βˆƒπ‘₯(𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑))))
196, 12, 18mp2and 698 1 (𝐽 ∈ Comp β†’ βˆƒπ‘₯(𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  βˆͺ cuni 4869  β€˜cfv 6500  Fincfn 8889  ficfi 9354  topGenctg 17327  Topctop 22265  Compccmp 22760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7807  df-1o 8416  df-er 8654  df-en 8890  df-fin 8893  df-fi 9355  df-topgen 17333  df-top 22266  df-cmp 22761
This theorem is referenced by:  alexsubALT  23425
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