Theorem conncompconn 22689

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 5005	. . . 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 2764	. . 3 ⊢ 𝑆 = ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}𝑦
4	3	oveq2i 7348	. 2 ⊢ (𝐽 ↾t 𝑆) = (𝐽 ↾t ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}𝑦)
5		simpl 483	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6		simpr 485	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
7		eleq2w 2820	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦))
8		oveq2 7345	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐽 ↾t 𝑥) = (𝐽 ↾t 𝑦))
9	8	eleq1d 2821	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝐽 ↾t 𝑥) ∈ Conn ↔ (𝐽 ↾t 𝑦) ∈ Conn))
10	7, 9	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn) ↔ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)))
11	10	elrab 3634	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn)))
12	6, 11	sylib 217	. . . . 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)))
13	12	simpld 495	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → 𝑦 ∈ 𝒫 𝑋)
14	13	elpwid 4556	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → 𝑦 ⊆ 𝑋)
15	12	simprd 496	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Conn))
16	15	simpld 495	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → 𝐴 ∈ 𝑦)
17	15	simprd 496	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}) → (𝐽 ↾t 𝑦) ∈ Conn)
18	5, 14, 16, 17	iunconn 22685	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}𝑦) ∈ Conn)
19	4, 18	eqeltrid 2841	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t 𝑆) ∈ Conn)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {crab 3403  𝒫 cpw 4547  ∪ cuni 4852  ∪ ciun 4941  ‘cfv 6479  (class class class)co 7337   ↾_t crest 17228  TopOnctopon 22165  Conncconn 22668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-en 8805  df-fin 8808  df-fi 9268  df-rest 17230  df-topgen 17251  df-top 22149  df-topon 22166  df-bases 22202  df-cld 22276  df-conn 22669

This theorem is referenced by:  conncompcld  22691  conncompclo  22692  tgpconncompeqg  23369  tgpconncomp  23370