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Mirrors > Home > MPE Home > Th. List > qtopcmplem | Structured version Visualization version GIF version |
Description: Lemma for qtopcmp 22244 and qtopconn 22245. (Contributed by Mario Carneiro, 24-Mar-2015.) |
Ref | Expression |
---|---|
qtopcmp.1 | ⊢ 𝑋 = ∪ 𝐽 |
qtopcmplem.1 | ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top) |
qtopcmplem.2 | ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ 𝐴) |
Ref | Expression |
---|---|
qtopcmplem | ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐽 ∈ 𝐴) | |
2 | simpr 485 | . . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹 Fn 𝑋) | |
3 | dffn4 6589 | . . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹) | |
4 | 2, 3 | sylib 219 | . . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋–onto→ran 𝐹) |
5 | qtopcmplem.1 | . . . . . 6 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top) | |
6 | qtopcmp.1 | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
7 | 6 | qtopuni 22238 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹)) |
8 | 5, 7 | sylan 580 | . . . . 5 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹)) |
9 | 3, 8 | sylan2b 593 | . . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → ran 𝐹 = ∪ (𝐽 qTop 𝐹)) |
10 | foeq3 6581 | . . . 4 ⊢ (ran 𝐹 = ∪ (𝐽 qTop 𝐹) → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹))) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐹:𝑋–onto→ran 𝐹 ↔ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹))) |
12 | 4, 11 | mpbid 233 | . 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹)) |
13 | 6 | toptopon 21453 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
14 | 5, 13 | sylib 219 | . . 3 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ (TopOn‘𝑋)) |
15 | qtopid 22241 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) | |
16 | 14, 15 | sylan 580 | . 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) |
17 | qtopcmplem.2 | . 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹) ∧ 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) → (𝐽 qTop 𝐹) ∈ 𝐴) | |
18 | 1, 12, 16, 17 | syl3anc 1363 | 1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∪ cuni 4830 ran crn 5549 Fn wfn 6343 –onto→wfo 6346 ‘cfv 6348 (class class class)co 7145 qTop cqtop 16764 Topctop 21429 TopOnctopon 21446 Cn ccn 21760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-qtop 16768 df-top 21430 df-topon 21447 df-cn 21763 |
This theorem is referenced by: qtopcmp 22244 qtopconn 22245 qtoppconn 32380 |
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