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Mirrors > Home > MPE Home > Th. List > qtopcmplem | Structured version Visualization version GIF version |
Description: Lemma for qtopcmp 21924 and qtopconn 21925. (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 476 | . 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐽 ∈ 𝐴) | |
2 | simpr 479 | . . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹 Fn 𝑋) | |
3 | dffn4 6374 | . . . 4 ⊢ (𝐹 Fn 𝑋 ↔ 𝐹:𝑋–onto→ran 𝐹) | |
4 | 2, 3 | sylib 210 | . . 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 21918 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹)) |
8 | 5, 7 | sylan 575 | . . . . 5 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹:𝑋–onto→ran 𝐹) → ran 𝐹 = ∪ (𝐽 qTop 𝐹)) |
9 | 3, 8 | sylan2b 587 | . . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → ran 𝐹 = ∪ (𝐽 qTop 𝐹)) |
10 | foeq3 6366 | . . . 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 224 | . 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → 𝐹:𝑋–onto→∪ (𝐽 qTop 𝐹)) |
13 | 6 | toptopon 21133 | . . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
14 | 5, 13 | sylib 210 | . . 3 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ (TopOn‘𝑋)) |
15 | qtopid 21921 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 Fn 𝑋) → 𝐹 ∈ (𝐽 Cn (𝐽 qTop 𝐹))) | |
16 | 14, 15 | sylan 575 | . 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 1439 | 1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐹 Fn 𝑋) → (𝐽 qTop 𝐹) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∪ cuni 4673 ran crn 5358 Fn wfn 6132 –onto→wfo 6135 ‘cfv 6137 (class class class)co 6924 qTop cqtop 16553 Topctop 21109 TopOnctopon 21126 Cn ccn 21440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-map 8144 df-qtop 16557 df-top 21110 df-topon 21127 df-cn 21443 |
This theorem is referenced by: qtopcmp 21924 qtopconn 21925 qtoppconn 31821 |
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