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Theorem qtopcmplem 23715
Description: Lemma for qtopcmp 23716 and qtopconn 23717. (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypotheses
Ref Expression
qtopcmp.1 𝑋 = 𝐽
qtopcmplem.1 (𝐽𝐴𝐽 ∈ Top)
qtopcmplem.2 ((𝐽𝐴𝐹:𝑋onto (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ 𝐴)
Assertion
Ref Expression
qtopcmplem ((𝐽𝐴𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)

Proof of Theorem qtopcmplem
StepHypRef Expression
1 simpl 482 . 2 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐽𝐴)
2 simpr 484 . . . 4 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹 Fn 𝑋)
3 dffn4 6826 . . . 4 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
42, 3sylib 218 . . 3 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹:𝑋onto→ran 𝐹)
5 qtopcmplem.1 . . . . . 6 (𝐽𝐴𝐽 ∈ Top)
6 qtopcmp.1 . . . . . . 7 𝑋 = 𝐽
76qtopuni 23710 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
85, 7sylan 580 . . . . 5 ((𝐽𝐴𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
93, 8sylan2b 594 . . . 4 ((𝐽𝐴𝐹 Fn 𝑋) → ran 𝐹 = (𝐽 qTop 𝐹))
10 foeq3 6818 . . . 4 (ran 𝐹 = (𝐽 qTop 𝐹) → (𝐹:𝑋onto→ran 𝐹𝐹:𝑋onto (𝐽 qTop 𝐹)))
119, 10syl 17 . . 3 ((𝐽𝐴𝐹 Fn 𝑋) → (𝐹:𝑋onto→ran 𝐹𝐹:𝑋onto (𝐽 qTop 𝐹)))
124, 11mpbid 232 . 2 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹:𝑋onto (𝐽 qTop 𝐹))
136toptopon 22923 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
145, 13sylib 218 . . 3 (𝐽𝐴𝐽 ∈ (TopOn‘𝑋))
15 qtopid 23713 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
1614, 15sylan 580 . 2 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
17 qtopcmplem.2 . 2 ((𝐽𝐴𝐹:𝑋onto (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ 𝐴)
181, 12, 16, 17syl3anc 1373 1 ((𝐽𝐴𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108   cuni 4907  ran crn 5686   Fn wfn 6556  ontowfo 6559  cfv 6561  (class class class)co 7431   qTop cqtop 17548  Topctop 22899  TopOnctopon 22916   Cn ccn 23232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-qtop 17552  df-top 22900  df-topon 22917  df-cn 23235
This theorem is referenced by:  qtopcmp  23716  qtopconn  23717  qtoppconn  35241
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