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Theorem qtopcmplem 23566
Description: Lemma for qtopcmp 23567 and qtopconn 23568. (Contributed by Mario Carneiro, 24-Mar-2015.)
Hypotheses
Ref Expression
qtopcmp.1 𝑋 = βˆͺ 𝐽
qtopcmplem.1 (𝐽 ∈ 𝐴 β†’ 𝐽 ∈ Top)
qtopcmplem.2 ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–ontoβ†’βˆͺ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) β†’ (𝐽 qTop 𝐹) ∈ 𝐴)
Assertion
Ref Expression
qtopcmplem ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ 𝐴)

Proof of Theorem qtopcmplem
StepHypRef Expression
1 simpl 482 . 2 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐽 ∈ 𝐴)
2 simpr 484 . . . 4 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐹 Fn 𝑋)
3 dffn4 6805 . . . 4 (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–ontoβ†’ran 𝐹)
42, 3sylib 217 . . 3 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐹:𝑋–ontoβ†’ran 𝐹)
5 qtopcmplem.1 . . . . . 6 (𝐽 ∈ 𝐴 β†’ 𝐽 ∈ Top)
6 qtopcmp.1 . . . . . . 7 𝑋 = βˆͺ 𝐽
76qtopuni 23561 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
85, 7sylan 579 . . . . 5 ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
93, 8sylan2b 593 . . . 4 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
10 foeq3 6797 . . . 4 (ran 𝐹 = βˆͺ (𝐽 qTop 𝐹) β†’ (𝐹:𝑋–ontoβ†’ran 𝐹 ↔ 𝐹:𝑋–ontoβ†’βˆͺ (𝐽 qTop 𝐹)))
119, 10syl 17 . . 3 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ (𝐹:𝑋–ontoβ†’ran 𝐹 ↔ 𝐹:𝑋–ontoβ†’βˆͺ (𝐽 qTop 𝐹)))
124, 11mpbid 231 . 2 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐹:𝑋–ontoβ†’βˆͺ (𝐽 qTop 𝐹))
136toptopon 22774 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
145, 13sylib 217 . . 3 (𝐽 ∈ 𝐴 β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
15 qtopid 23564 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
1614, 15sylan 579 . 2 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
17 qtopcmplem.2 . 2 ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–ontoβ†’βˆͺ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) β†’ (𝐽 qTop 𝐹) ∈ 𝐴)
181, 12, 16, 17syl3anc 1368 1 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ (𝐽 qTop 𝐹) ∈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆͺ cuni 4902  ran crn 5670   Fn wfn 6532  β€“ontoβ†’wfo 6535  β€˜cfv 6537  (class class class)co 7405   qTop cqtop 17458  Topctop 22750  TopOnctopon 22767   Cn ccn 23083
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824  df-qtop 17462  df-top 22751  df-topon 22768  df-cn 23086
This theorem is referenced by:  qtopcmp  23567  qtopconn  23568  qtoppconn  34755
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