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Mirrors > Home > MPE Home > Th. List > qtopcmplem | Structured version Visualization version GIF version |
Description: Lemma for qtopcmp 23082 and qtopconn 23083. (Contributed by Mario Carneiro, 24-Mar-2015.) |
Ref | Expression |
---|---|
qtopcmp.1 | β’ π = βͺ π½ |
qtopcmplem.1 | β’ (π½ β π΄ β π½ β Top) |
qtopcmplem.2 | β’ ((π½ β π΄ β§ πΉ:πβontoββͺ (π½ qTop πΉ) β§ πΉ β (π½ Cn (π½ qTop πΉ))) β (π½ qTop πΉ) β π΄) |
Ref | Expression |
---|---|
qtopcmplem | β’ ((π½ β π΄ β§ πΉ Fn π) β (π½ qTop πΉ) β π΄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . 2 β’ ((π½ β π΄ β§ πΉ Fn π) β π½ β π΄) | |
2 | simpr 486 | . . . 4 β’ ((π½ β π΄ β§ πΉ Fn π) β πΉ Fn π) | |
3 | dffn4 6766 | . . . 4 β’ (πΉ Fn π β πΉ:πβontoβran πΉ) | |
4 | 2, 3 | sylib 217 | . . 3 β’ ((π½ β π΄ β§ πΉ Fn π) β πΉ:πβontoβran πΉ) |
5 | qtopcmplem.1 | . . . . . 6 β’ (π½ β π΄ β π½ β Top) | |
6 | qtopcmp.1 | . . . . . . 7 β’ π = βͺ π½ | |
7 | 6 | qtopuni 23076 | . . . . . 6 β’ ((π½ β Top β§ πΉ:πβontoβran πΉ) β ran πΉ = βͺ (π½ qTop πΉ)) |
8 | 5, 7 | sylan 581 | . . . . 5 β’ ((π½ β π΄ β§ πΉ:πβontoβran πΉ) β ran πΉ = βͺ (π½ qTop πΉ)) |
9 | 3, 8 | sylan2b 595 | . . . 4 β’ ((π½ β π΄ β§ πΉ Fn π) β ran πΉ = βͺ (π½ qTop πΉ)) |
10 | foeq3 6758 | . . . 4 β’ (ran πΉ = βͺ (π½ qTop πΉ) β (πΉ:πβontoβran πΉ β πΉ:πβontoββͺ (π½ qTop πΉ))) | |
11 | 9, 10 | syl 17 | . . 3 β’ ((π½ β π΄ β§ πΉ Fn π) β (πΉ:πβontoβran πΉ β πΉ:πβontoββͺ (π½ qTop πΉ))) |
12 | 4, 11 | mpbid 231 | . 2 β’ ((π½ β π΄ β§ πΉ Fn π) β πΉ:πβontoββͺ (π½ qTop πΉ)) |
13 | 6 | toptopon 22289 | . . . 4 β’ (π½ β Top β π½ β (TopOnβπ)) |
14 | 5, 13 | sylib 217 | . . 3 β’ (π½ β π΄ β π½ β (TopOnβπ)) |
15 | qtopid 23079 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β πΉ β (π½ Cn (π½ qTop πΉ))) | |
16 | 14, 15 | sylan 581 | . 2 β’ ((π½ β π΄ β§ πΉ Fn π) β πΉ β (π½ Cn (π½ qTop πΉ))) |
17 | qtopcmplem.2 | . 2 β’ ((π½ β π΄ β§ πΉ:πβontoββͺ (π½ qTop πΉ) β§ πΉ β (π½ Cn (π½ qTop πΉ))) β (π½ qTop πΉ) β π΄) | |
18 | 1, 12, 16, 17 | syl3anc 1372 | 1 β’ ((π½ β π΄ β§ πΉ Fn π) β (π½ qTop πΉ) β π΄) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 βͺ cuni 4869 ran crn 5638 Fn wfn 6495 βontoβwfo 6498 βcfv 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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