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Theorem qtopcmplem 21420
Description: Lemma for qtopcmp 21421 and qtopconn 21422. (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 473 . 2 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐽𝐴)
2 simpr 477 . . . 4 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹 Fn 𝑋)
3 dffn4 6078 . . . 4 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
42, 3sylib 208 . . 3 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹:𝑋onto→ran 𝐹)
5 qtopcmplem.1 . . . . . 6 (𝐽𝐴𝐽 ∈ Top)
6 qtopcmp.1 . . . . . . 7 𝑋 = 𝐽
76qtopuni 21415 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
85, 7sylan 488 . . . . 5 ((𝐽𝐴𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
93, 8sylan2b 492 . . . 4 ((𝐽𝐴𝐹 Fn 𝑋) → ran 𝐹 = (𝐽 qTop 𝐹))
10 foeq3 6070 . . . 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 222 . 2 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹:𝑋onto (𝐽 qTop 𝐹))
136toptopon 20648 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
145, 13sylib 208 . . 3 (𝐽𝐴𝐽 ∈ (TopOn‘𝑋))
15 qtopid 21418 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
1614, 15sylan 488 . 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 1323 1 ((𝐽𝐴𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987   cuni 4402  ran crn 5075   Fn wfn 5842  ontowfo 5845  cfv 5847  (class class class)co 6604   qTop cqtop 16084  Topctop 20617  TopOnctopon 20618   Cn ccn 20938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-qtop 16088  df-top 20621  df-topon 20623  df-cn 20941
This theorem is referenced by:  qtopcmp  21421  qtopconn  21422  qtoppconn  30926
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