Theorem qtopcmplem 23668

Description: Lemma for qtopcmp 23669 and qtopconn 23670. (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 482	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐽 ∈ 𝐴)
2		simpr 484	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹 Fn 𝑋)
3		dffn4 6762	. . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
4	2, 3	sylib 218	. . 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 23663	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
8	5, 7	sylan 581	. . . . 5 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
9	3, 8	sylan2b 595	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
10		foeq3 6754	. . . 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 232	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹))
13	6	toptopon 22878	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
14	5, 13	sylib 218	. . 3 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ (TopOn‘𝑋))
15		qtopid 23666	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
16	14, 15	sylan 581	. 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 1374	1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∪ cuni 4865  ran crn 5635   Fn wfn 6497  –onto→wfo 6500  ‘cfv 6502  (class class class)co 7370   qTop cqtop 17438  Topctop 22854  TopOnctopon 22871   Cn ccn 23185

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-map 8779  df-qtop 17442  df-top 22855  df-topon 22872  df-cn 23188

This theorem is referenced by:  qtopcmp  23669  qtopconn  23670  qtoppconn  35458