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Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem2 | Structured version Visualization version GIF version |
Description: Lemma for heibor 37145. Substitutions for the set πΊ. (Contributed by Jeff Madsen, 23-Jan-2014.) |
Ref | Expression |
---|---|
heibor.1 | β’ π½ = (MetOpenβπ·) |
heibor.3 | β’ πΎ = {π’ β£ Β¬ βπ£ β (π« π β© Fin)π’ β βͺ π£} |
heibor.4 | β’ πΊ = {β¨π¦, πβ© β£ (π β β0 β§ π¦ β (πΉβπ) β§ (π¦π΅π) β πΎ)} |
heiborlem2.5 | β’ π΄ β V |
heiborlem2.6 | β’ πΆ β V |
Ref | Expression |
---|---|
heiborlem2 | β’ (π΄πΊπΆ β (πΆ β β0 β§ π΄ β (πΉβπΆ) β§ (π΄π΅πΆ) β πΎ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | heiborlem2.5 | . 2 β’ π΄ β V | |
2 | heiborlem2.6 | . 2 β’ πΆ β V | |
3 | eleq1 2813 | . . 3 β’ (π¦ = π΄ β (π¦ β (πΉβπ) β π΄ β (πΉβπ))) | |
4 | oveq1 7408 | . . . 4 β’ (π¦ = π΄ β (π¦π΅π) = (π΄π΅π)) | |
5 | 4 | eleq1d 2810 | . . 3 β’ (π¦ = π΄ β ((π¦π΅π) β πΎ β (π΄π΅π) β πΎ)) |
6 | 3, 5 | 3anbi23d 1435 | . 2 β’ (π¦ = π΄ β ((π β β0 β§ π¦ β (πΉβπ) β§ (π¦π΅π) β πΎ) β (π β β0 β§ π΄ β (πΉβπ) β§ (π΄π΅π) β πΎ))) |
7 | eleq1 2813 | . . 3 β’ (π = πΆ β (π β β0 β πΆ β β0)) | |
8 | fveq2 6881 | . . . 4 β’ (π = πΆ β (πΉβπ) = (πΉβπΆ)) | |
9 | 8 | eleq2d 2811 | . . 3 β’ (π = πΆ β (π΄ β (πΉβπ) β π΄ β (πΉβπΆ))) |
10 | oveq2 7409 | . . . 4 β’ (π = πΆ β (π΄π΅π) = (π΄π΅πΆ)) | |
11 | 10 | eleq1d 2810 | . . 3 β’ (π = πΆ β ((π΄π΅π) β πΎ β (π΄π΅πΆ) β πΎ)) |
12 | 7, 9, 11 | 3anbi123d 1432 | . 2 β’ (π = πΆ β ((π β β0 β§ π΄ β (πΉβπ) β§ (π΄π΅π) β πΎ) β (πΆ β β0 β§ π΄ β (πΉβπΆ) β§ (π΄π΅πΆ) β πΎ))) |
13 | heibor.4 | . 2 β’ πΊ = {β¨π¦, πβ© β£ (π β β0 β§ π¦ β (πΉβπ) β§ (π¦π΅π) β πΎ)} | |
14 | 1, 2, 6, 12, 13 | brab 5533 | 1 β’ (π΄πΊπΆ β (πΆ β β0 β§ π΄ β (πΉβπΆ) β§ (π΄π΅πΆ) β πΎ)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wb 205 β§ w3a 1084 = wceq 1533 β wcel 2098 {cab 2701 βwrex 3062 Vcvv 3466 β© cin 3939 β wss 3940 π« cpw 4594 βͺ cuni 4899 class class class wbr 5138 {copab 5200 βcfv 6533 (class class class)co 7401 Fincfn 8934 β0cn0 12468 MetOpencmopn 21217 |
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-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
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-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-iota 6485 df-fv 6541 df-ov 7404 |
This theorem is referenced by: heiborlem3 37137 heiborlem5 37139 heiborlem6 37140 heiborlem8 37142 heiborlem10 37144 |
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