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Theorem qtopcmplem 22318
Description: Lemma for qtopcmp 22319 and qtopconn 22320. (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 486 . 2 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐽𝐴)
2 simpr 488 . . . 4 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹 Fn 𝑋)
3 dffn4 6587 . . . 4 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
42, 3sylib 221 . . 3 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹:𝑋onto→ran 𝐹)
5 qtopcmplem.1 . . . . . 6 (𝐽𝐴𝐽 ∈ Top)
6 qtopcmp.1 . . . . . . 7 𝑋 = 𝐽
76qtopuni 22313 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
85, 7sylan 583 . . . . 5 ((𝐽𝐴𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
93, 8sylan2b 596 . . . 4 ((𝐽𝐴𝐹 Fn 𝑋) → ran 𝐹 = (𝐽 qTop 𝐹))
10 foeq3 6579 . . . 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 235 . 2 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹:𝑋onto (𝐽 qTop 𝐹))
136toptopon 21528 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
145, 13sylib 221 . . 3 (𝐽𝐴𝐽 ∈ (TopOn‘𝑋))
15 qtopid 22316 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
1614, 15sylan 583 . 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 1368 1 ((𝐽𝐴𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115   cuni 4824  ran crn 5543   Fn wfn 6338  ontowfo 6341  cfv 6343  (class class class)co 7149   qTop cqtop 16776  Topctop 21504  TopOnctopon 21521   Cn ccn 21835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404  df-qtop 16780  df-top 21505  df-topon 21522  df-cn 21838
This theorem is referenced by:  qtopcmp  22319  qtopconn  22320  qtoppconn  32543
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