Theorem qtopcmplem 23210

Description: Lemma for qtopcmp 23211 and qtopconn 23212. (Contributed by Mario Carneiro, 24-Mar-2015.)

Hypotheses

Ref	Expression
qtopcmp.1	⊢ 𝑋 = ∪ 𝐽
qtopcmplem.1	⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
qtopcmplem.2	⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ 𝐴)

Assertion

Ref	Expression
qtopcmplem	⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)

Proof of Theorem qtopcmplem

Step	Hyp	Ref	Expression
1		simpl 483	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐽 ∈ 𝐴)
2		simpr 485	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹 Fn 𝑋)
3		dffn4 6811	. . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
4	2, 3	sylib 217	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋–onto→ran 𝐹)
5		qtopcmplem.1	. . . . . 6 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
6		qtopcmp.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
7	6	qtopuni 23205	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
8	5, 7	sylan 580	. . . . 5 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
9	3, 8	sylan2b 594	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
10		foeq3 6803	. . . 4 ⊢ (ran 𝐹 = ∪ (𝐽 qTop 𝐹) → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹)))
11	9, 10	syl 17	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹)))
12	4, 11	mpbid 231	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹))
13	6	toptopon 22418	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
14	5, 13	sylib 217	. . 3 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ (TopOn‘𝑋))
15		qtopid 23208	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
16	14, 15	sylan 580	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
17		qtopcmplem.2	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ 𝐴)
18	1, 12, 16, 17	syl3anc 1371	1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∪ cuni 4908  ran crn 5677   Fn wfn 6538  –onto→wfo 6541  ‘cfv 6543  (class class class)co 7408   qTop cqtop 17448  Topctop 22394  TopOnctopon 22411   Cn ccn 22727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-qtop 17452  df-top 22395  df-topon 22412  df-cn 22730

This theorem is referenced by:  qtopcmp  23211  qtopconn  23212  qtoppconn  34222