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Theorem qtopcmplem 21712
 Description: Lemma for qtopcmp 21713 and qtopconn 21714. (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 474 . 2 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐽𝐴)
2 simpr 479 . . . 4 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹 Fn 𝑋)
3 dffn4 6282 . . . 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 21707 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
85, 7sylan 489 . . . . 5 ((𝐽𝐴𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
93, 8sylan2b 493 . . . 4 ((𝐽𝐴𝐹 Fn 𝑋) → ran 𝐹 = (𝐽 qTop 𝐹))
10 foeq3 6274 . . . 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 20924 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
145, 13sylib 208 . . 3 (𝐽𝐴𝐽 ∈ (TopOn‘𝑋))
15 qtopid 21710 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
1614, 15sylan 489 . 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 1477 1 ((𝐽𝐴𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∪ cuni 4588  ran crn 5267   Fn wfn 6044  –onto→wfo 6047  ‘cfv 6049  (class class class)co 6813   qTop cqtop 16365  Topctop 20900  TopOnctopon 20917   Cn ccn 21230 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-qtop 16369  df-top 20901  df-topon 20918  df-cn 21233 This theorem is referenced by:  qtopcmp  21713  qtopconn  21714  qtoppconn  31525
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