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Theorem 1stcclb 21467
Description: A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
Hypothesis
Ref Expression
1stcclb.1 𝑋 = 𝐽
Assertion
Ref Expression
1stcclb ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐽,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem 1stcclb
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 1stcclb.1 . . . 4 𝑋 = 𝐽
21is1stc2 21465 . . 3 (𝐽 ∈ 1st𝜔 ↔ (𝐽 ∈ Top ∧ ∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)))))
32simprbi 478 . 2 (𝐽 ∈ 1st𝜔 → ∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))))
4 eleq1 2837 . . . . . . 7 (𝑤 = 𝐴 → (𝑤𝑦𝐴𝑦))
5 eleq1 2837 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑤𝑧𝐴𝑧))
65anbi1d 607 . . . . . . . 8 (𝑤 = 𝐴 → ((𝑤𝑧𝑧𝑦) ↔ (𝐴𝑧𝑧𝑦)))
76rexbidv 3199 . . . . . . 7 (𝑤 = 𝐴 → (∃𝑧𝑥 (𝑤𝑧𝑧𝑦) ↔ ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))
84, 7imbi12d 333 . . . . . 6 (𝑤 = 𝐴 → ((𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)) ↔ (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
98ralbidv 3134 . . . . 5 (𝑤 = 𝐴 → (∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)) ↔ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
109anbi2d 606 . . . 4 (𝑤 = 𝐴 → ((𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) ↔ (𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
1110rexbidv 3199 . . 3 (𝑤 = 𝐴 → (∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) ↔ ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
1211rspcv 3454 . 2 (𝐴𝑋 → (∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
133, 12mpan9 490 1 ((𝐽 ∈ 1st𝜔 ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061  wss 3721  𝒫 cpw 4295   cuni 4572   class class class wbr 4784  ωcom 7211  cdom 8106  Topctop 20917  1st𝜔c1stc 21460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-in 3728  df-ss 3735  df-pw 4297  df-uni 4573  df-1stc 21462
This theorem is referenced by:  1stcfb  21468  1stcrest  21476  lly1stc  21519  tx1stc  21673
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