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Theorem qtopcmplem 23629
Description: Lemma for qtopcmp 23630 and qtopconn 23631. (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypotheses
Ref Expression
qtopcmp.1 𝑋 = βˆͺ 𝐽
qtopcmplem.1 (𝐽 ∈ 𝐴 β†’ 𝐽 ∈ Top)
qtopcmplem.2 ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–ontoβ†’βˆͺ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) β†’ (𝐽 qTop 𝐹) ∈ 𝐴)
Assertion
Ref Expression
qtopcmplem ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ 𝐴)

Proof of Theorem qtopcmplem
StepHypRef Expression
1 simpl 481 . 2 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐽 ∈ 𝐴)
2 simpr 483 . . . 4 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐹 Fn 𝑋)
3 dffn4 6812 . . . 4 (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–ontoβ†’ran 𝐹)
42, 3sylib 217 . . 3 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐹:𝑋–ontoβ†’ran 𝐹)
5 qtopcmplem.1 . . . . . 6 (𝐽 ∈ 𝐴 β†’ 𝐽 ∈ Top)
6 qtopcmp.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
76qtopuni 23624 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
85, 7sylan 578 . . . . 5 ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
93, 8sylan2b 592 . . . 4 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
10 foeq3 6804 . . . 4 (ran 𝐹 = βˆͺ (𝐽 qTop 𝐹) β†’ (𝐹:𝑋–ontoβ†’ran 𝐹 ↔ 𝐹:𝑋–ontoβ†’βˆͺ (𝐽 qTop 𝐹)))
119, 10syl 17 . . 3 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ (𝐹:𝑋–ontoβ†’ran 𝐹 ↔ 𝐹:𝑋–ontoβ†’βˆͺ (𝐽 qTop 𝐹)))
124, 11mpbid 231 . 2 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐹:𝑋–ontoβ†’βˆͺ (𝐽 qTop 𝐹))
136toptopon 22837 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
145, 13sylib 217 . . 3 (𝐽 ∈ 𝐴 β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
15 qtopid 23627 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
1614, 15sylan 578 . 2 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
17 qtopcmplem.2 . 2 ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–ontoβ†’βˆͺ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) β†’ (𝐽 qTop 𝐹) ∈ 𝐴)
181, 12, 16, 17syl3anc 1368 1 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆͺ cuni 4903  ran crn 5673   Fn wfn 6538  β€“ontoβ†’wfo 6541  β€˜cfv 6543  (class class class)co 7416   qTop cqtop 17484  Topctop 22813  TopOnctopon 22830   Cn ccn 23146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7419  df-oprab 7420  df-mpo 7421  df-map 8845  df-qtop 17488  df-top 22814  df-topon 22831  df-cn 23149
This theorem is referenced by:  qtopcmp  23630  qtopconn  23631  qtoppconn  34903
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