Theorem qtopcmplem 24053

Description: Lemma for qtopcmp 24054 and qtopconn 24055. (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 488	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐽 ∈ 𝐴)
2		dffn4 6821	. . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋⟶ontoran 𝐹)
3	2	bilani 510	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋⟶ontoran 𝐹)
4		qtopcmplem.1	. . . . . 6 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
5		qtopcmp.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
6	5	qtopuni 24048	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋⟶ontoran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
7	4, 6	sylan 592	. . . . 5 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋⟶ontoran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
8	2, 7	sylan2b 606	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
9		foeq3 6813	. . . 4 ⊢ (ran 𝐹 = ∪ (𝐽 qTop 𝐹) → (𝐹:𝑋⟶ontoran 𝐹 ↔ 𝐹:𝑋⟶onto∪ (𝐽 qTop 𝐹)))
10	8, 9	syl 18	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐹:𝑋⟶ontoran 𝐹 ↔ 𝐹:𝑋⟶onto∪ (𝐽 qTop 𝐹)))
11	3, 10	mpbid 235	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋⟶onto∪ (𝐽 qTop 𝐹))
12	5	toptopon 23262	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
13	4, 12	sylib 221	. . 3 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ (TopOn‘𝑋))
14		qtopid 24051	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
15	13, 14	sylan 592	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
16		qtopcmplem.2	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋⟶onto∪ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ 𝐴)
17	1, 11, 15, 16	syl3anc 1399	1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∪ cuni 4870  ran crn 5656   Fn wfn 6550  ⟶ontowfo 6553  ‘cfv 6555  (class class class)co 7440   qTop cqtop 17701  Topctop 23238  TopOnctopon 23255   Cn ccn 23569

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 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-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-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-map 8874  df-qtop 17705  df-top 23239  df-topon 23256  df-cn 23572

This theorem is used by:  qtopcmp  24054  qtopconn  24055  qtoppconn  36167