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Theorem qtopcmplem 23598
Description: Lemma for qtopcmp 23599 and qtopconn 23600. (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 6811 . . . 4 (𝐹 Fn 𝑋𝐹:𝑋onto→ran 𝐹)
42, 3sylib 217 . . 3 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹:𝑋onto→ran 𝐹)
5 qtopcmplem.1 . . . . . 6 (𝐽𝐴𝐽 ∈ Top)
6 qtopcmp.1 . . . . . . 7 𝑋 = 𝐽
76qtopuni 23593 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
85, 7sylan 579 . . . . 5 ((𝐽𝐴𝐹:𝑋onto→ran 𝐹) → ran 𝐹 = (𝐽 qTop 𝐹))
93, 8sylan2b 593 . . . 4 ((𝐽𝐴𝐹 Fn 𝑋) → ran 𝐹 = (𝐽 qTop 𝐹))
10 foeq3 6803 . . . 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 231 . 2 ((𝐽𝐴𝐹 Fn 𝑋) → 𝐹:𝑋onto (𝐽 qTop 𝐹))
136toptopon 22806 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
145, 13sylib 217 . . 3 (𝐽𝐴𝐽 ∈ (TopOn‘𝑋))
15 qtopid 23596 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
1614, 15sylan 579 . 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 1369 1 ((𝐽𝐴𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085   = wceq 1534  wcel 2099   cuni 4903  ran crn 5673   Fn wfn 6537  ontowfo 6540  cfv 6542  (class class class)co 7414   qTop cqtop 17476  Topctop 22782  TopOnctopon 22799   Cn ccn 23115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  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 5143  df-opab 5205  df-mpt 5226  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 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8838  df-qtop 17480  df-top 22783  df-topon 22800  df-cn 23118
This theorem is referenced by:  qtopcmp  23599  qtopconn  23600  qtoppconn  34782
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