Theorem conncompconn 23783

Description: The connected component containing 𝐴 is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypothesis

Ref	Expression
conncomp.2	⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}

Assertion

Ref	Expression
conncompconn	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t 𝑆) ∈ Conn)

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

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem conncompconn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		conncomp.2	. . . 4 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}
2		uniiun 5020	. . . 4 ⊢ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} = ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}𝑦
3	1, 2	eqtri 2786	. . 3 ⊢ 𝑆 = ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}𝑦
4	3	oveq2i 7457	. 2 ⊢ (𝐽 ↾t 𝑆) = (𝐽 ↾t ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}𝑦)
5		simpl 488	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6		eleq2w 2848	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
7		oveq2 7454	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐽 ↾t 𝑥) = (𝐽 ↾t 𝑦))
8	7	eleq1d 2849	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐽 ↾t 𝑥) ∈ Conn ↔ (𝐽 ↾t 𝑦) ∈ Conn))
9	6, 8	anbi12d 644	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn) ↔ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)))
10	9	elrab 3648	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)))
11	10	bilani 510	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)))
12	11	simpld 500	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → 𝑦 ∈ 𝒫 𝑋)
13	12	elpwid 4569	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → 𝑦 ⊆ 𝑋)
14	11	simprd 501	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn))
15	14	simpld 500	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → 𝐴 ∈ 𝑦)
16	14	simprd 501	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → (𝐽 ↾t 𝑦) ∈ Conn)
17	5, 13, 15, 16	iunconn 23779	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}𝑦) ∈ Conn)
18	4, 17	eqeltrid 2868	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t 𝑆) ∈ Conn)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  𝒫 cpw 4560  ∪ cuni 4870  ∪ ciun 4954  ‘cfv 6561  (class class class)co 7446   ↾_t crest 17623  TopOnctopon 23261  Conncconn 23762

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-en 9005  df-fin 9008  df-fi 9433  df-rest 17625  df-topgen 17646  df-top 23245  df-topon 23262  df-bases 23297  df-cld 23370  df-conn 23763

This theorem is used by:  conncompcld  23785  conncompclo  23786  tgpconncompeqg  24464  tgpconncomp  24465