Theorem qtopcmplem 23875

Description: Lemma for qtopcmp 23876 and qtopconn 23877. (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 487	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐽 ∈ 𝐴)
2		dffn4 6798	. . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹)
3	2	bilani 509	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋–onto→ran 𝐹)
4		qtopcmplem.1	. . . . . 6 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
5		qtopcmp.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
6	5	qtopuni 23870	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
7	4, 6	sylan 591	. . . . 5 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
8	2, 7	sylan2b 605	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → ran 𝐹 = ∪ (𝐽 qTop 𝐹))
9		foeq3 6790	. . . 4 ⊢ (ran 𝐹 = ∪ (𝐽 qTop 𝐹) → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹)))
10	8, 9	syl 18	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐹:𝑋–onto→ran 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 23085	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
13	4, 12	sylib 221	. . 3 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ (TopOn‘𝑋))
14		qtopid 23873	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
15	13, 14	sylan 591	. 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 1397	1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∪ cuni 4871  ran crn 5661   Fn wfn 6531  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7412   qTop cqtop 17563  Topctop 23061  TopOnctopon 23078   Cn ccn 23392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8824  df-qtop 17567  df-top 23062  df-topon 23079  df-cn 23395

This theorem is used by:  qtopcmp  23876  qtopconn  23877  qtoppconn  35736