Theorem opncldf2 22979

Description: The values of the open-closed bijection. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
opncldf.1	⊢ 𝑋 = ∪ 𝐽
opncldf.2	⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))

Assertion

Ref	Expression
opncldf2	⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐹‘𝐴) = (𝑋 ∖ 𝐴))

Distinct variable groups:   𝑢,𝐴   𝑢,𝐽   𝑢,𝑋

Allowed substitution hint:   𝐹(𝑢)

Proof of Theorem opncldf2

Step	Hyp	Ref	Expression
1		opncldf.2	. 2 ⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))
2		difeq2 4086	. 2 ⊢ (𝑢 = 𝐴 → (𝑋 ∖ 𝑢) = (𝑋 ∖ 𝐴))
3		simpr 484	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ∈ 𝐽)
4		opncldf.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
5	4	opncld 22927	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝑋 ∖ 𝐴) ∈ (Clsd‘𝐽))
6	1, 2, 3, 5	fvmptd3 6994	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → (𝐹‘𝐴) = (𝑋 ∖ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∖ cdif 3914  ∪ cuni 4874   ↦ cmpt 5191  ‘cfv 6514  Topctop 22787  Clsdccld 22910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-top 22788  df-cld 22913

This theorem is referenced by: (None)