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Mirrors > Home > MPE Home > Th. List > hsmexlem6 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 10369. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
Ref | Expression |
---|---|
hsmexlem4.x | β’ π β V |
hsmexlem4.h | β’ π» = (rec((π§ β V β¦ (harβπ« (π Γ π§))), (harβπ« π)) βΎ Ο) |
hsmexlem4.u | β’ π = (π₯ β V β¦ (rec((π¦ β V β¦ βͺ π¦), π₯) βΎ Ο)) |
hsmexlem4.s | β’ π = {π β βͺ (π 1 β On) β£ βπ β (TCβ{π})π βΌ π} |
hsmexlem4.o | β’ π = OrdIso( E , (rank β ((πβπ)βπ))) |
Ref | Expression |
---|---|
hsmexlem6 | β’ π β V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6856 | . 2 β’ (π 1β(harβπ« (Ο Γ βͺ ran π»))) β V | |
2 | hsmexlem4.x | . . . . 5 β’ π β V | |
3 | hsmexlem4.h | . . . . 5 β’ π» = (rec((π§ β V β¦ (harβπ« (π Γ π§))), (harβπ« π)) βΎ Ο) | |
4 | hsmexlem4.u | . . . . 5 β’ π = (π₯ β V β¦ (rec((π¦ β V β¦ βͺ π¦), π₯) βΎ Ο)) | |
5 | hsmexlem4.s | . . . . 5 β’ π = {π β βͺ (π 1 β On) β£ βπ β (TCβ{π})π βΌ π} | |
6 | hsmexlem4.o | . . . . 5 β’ π = OrdIso( E , (rank β ((πβπ)βπ))) | |
7 | 2, 3, 4, 5, 6 | hsmexlem5 10367 | . . . 4 β’ (π β π β (rankβπ) β (harβπ« (Ο Γ βͺ ran π»))) |
8 | 5 | ssrab3 4041 | . . . . . 6 β’ π β βͺ (π 1 β On) |
9 | 8 | sseli 3941 | . . . . 5 β’ (π β π β π β βͺ (π 1 β On)) |
10 | harcl 9496 | . . . . . 6 β’ (harβπ« (Ο Γ βͺ ran π»)) β On | |
11 | r1fnon 9704 | . . . . . . 7 β’ π 1 Fn On | |
12 | 11 | fndmi 6607 | . . . . . 6 β’ dom π 1 = On |
13 | 10, 12 | eleqtrri 2837 | . . . . 5 β’ (harβπ« (Ο Γ βͺ ran π»)) β dom π 1 |
14 | rankr1ag 9739 | . . . . 5 β’ ((π β βͺ (π 1 β On) β§ (harβπ« (Ο Γ βͺ ran π»)) β dom π 1) β (π β (π 1β(harβπ« (Ο Γ βͺ ran π»))) β (rankβπ) β (harβπ« (Ο Γ βͺ ran π»)))) | |
15 | 9, 13, 14 | sylancl 587 | . . . 4 β’ (π β π β (π β (π 1β(harβπ« (Ο Γ βͺ ran π»))) β (rankβπ) β (harβπ« (Ο Γ βͺ ran π»)))) |
16 | 7, 15 | mpbird 257 | . . 3 β’ (π β π β π β (π 1β(harβπ« (Ο Γ βͺ ran π»)))) |
17 | 16 | ssriv 3949 | . 2 β’ π β (π 1β(harβπ« (Ο Γ βͺ ran π»))) |
18 | 1, 17 | ssexi 5280 | 1 β’ π β V |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1542 β wcel 2107 βwral 3065 {crab 3408 Vcvv 3446 π« cpw 4561 {csn 4587 βͺ cuni 4866 class class class wbr 5106 β¦ cmpt 5189 E cep 5537 Γ cxp 5632 dom cdm 5634 ran crn 5635 βΎ cres 5636 β cima 5637 Oncon0 6318 βcfv 6497 Οcom 7803 reccrdg 8356 βΌ cdom 8882 OrdIsocoi 9446 harchar 9493 TCctc 9673 π 1cr1 9699 rankcrnk 9700 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-smo 8293 df-recs 8318 df-rdg 8357 df-en 8885 df-dom 8886 df-sdom 8887 df-oi 9447 df-har 9494 df-wdom 9502 df-tc 9674 df-r1 9701 df-rank 9702 |
This theorem is referenced by: hsmex 10369 |
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