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Theorem qtopcmplem 23081
Description: Lemma for qtopcmp 23082 and qtopconn 23083. (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 484 . 2 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐽 ∈ 𝐴)
2 simpr 486 . . . 4 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ 𝐹 Fn 𝑋)
3 dffn4 6766 . . . 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 23076 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
85, 7sylan 581 . . . . 5 ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–ontoβ†’ran 𝐹) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
93, 8sylan2b 595 . . . 4 ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) β†’ ran 𝐹 = βˆͺ (𝐽 qTop 𝐹))
10 foeq3 6758 . . . 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 22289 . . . 4 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜π‘‹))
145, 13sylib 217 . . 3 (𝐽 ∈ 𝐴 β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
15 qtopid 23079 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐹 Fn 𝑋) β†’ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹)))
1614, 15sylan 581 . 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 1372 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆͺ cuni 4869  ran crn 5638   Fn wfn 6495  β€“ontoβ†’wfo 6498  β€˜cfv 6500  (class class class)co 7361   qTop cqtop 17393  Topctop 22265  TopOnctopon 22282   Cn ccn 22598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773  df-qtop 17397  df-top 22266  df-topon 22283  df-cn 22601
This theorem is referenced by:  qtopcmp  23082  qtopconn  23083  qtoppconn  33894
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