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Mirrors > Home > MPE Home > Th. List > qtopcmplem | Structured version Visualization version GIF version |
Description: Lemma for qtopcmp 23567 and qtopconn 23568. (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 482 | . 2 β’ ((π½ β π΄ β§ πΉ Fn π) β π½ β π΄) | |
2 | simpr 484 | . . . 4 β’ ((π½ β π΄ β§ πΉ Fn π) β πΉ Fn π) | |
3 | dffn4 6805 | . . . 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 23561 | . . . . . 6 β’ ((π½ β Top β§ πΉ:πβontoβran πΉ) β ran πΉ = βͺ (π½ qTop πΉ)) |
8 | 5, 7 | sylan 579 | . . . . 5 β’ ((π½ β π΄ β§ πΉ:πβontoβran πΉ) β ran πΉ = βͺ (π½ qTop πΉ)) |
9 | 3, 8 | sylan2b 593 | . . . 4 β’ ((π½ β π΄ β§ πΉ Fn π) β ran πΉ = βͺ (π½ qTop πΉ)) |
10 | foeq3 6797 | . . . 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 22774 | . . . 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 23564 | . . 3 β’ ((π½ β (TopOnβπ) β§ πΉ Fn π) β πΉ β (π½ Cn (π½ qTop πΉ))) | |
16 | 14, 15 | sylan 579 | . 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 1368 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βͺ cuni 4902 ran crn 5670 Fn wfn 6532 βontoβwfo 6535 βcfv 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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