![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > hsmexlem6 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 10455. (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 6905 | . 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 10453 | . . . 4 β’ (π β π β (rankβπ) β (harβπ« (Ο Γ βͺ ran π»))) |
8 | 5 | ssrab3 4072 | . . . . . 6 β’ π β βͺ (π 1 β On) |
9 | 8 | sseli 3968 | . . . . 5 β’ (π β π β π β βͺ (π 1 β On)) |
10 | harcl 9582 | . . . . . 6 β’ (harβπ« (Ο Γ βͺ ran π»)) β On | |
11 | r1fnon 9790 | . . . . . . 7 β’ π 1 Fn On | |
12 | 11 | fndmi 6653 | . . . . . 6 β’ dom π 1 = On |
13 | 10, 12 | eleqtrri 2824 | . . . . 5 β’ (harβπ« (Ο Γ βͺ ran π»)) β dom π 1 |
14 | rankr1ag 9825 | . . . . 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 584 | . . . 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 256 | . . 3 β’ (π β π β π β (π 1β(harβπ« (Ο Γ βͺ ran π»)))) |
17 | 16 | ssriv 3976 | . 2 β’ π β (π 1β(harβπ« (Ο Γ βͺ ran π»))) |
18 | 1, 17 | ssexi 5317 | 1 β’ π β V |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1533 β wcel 2098 βwral 3051 {crab 3419 Vcvv 3463 π« cpw 4598 {csn 4624 βͺ cuni 4903 class class class wbr 5143 β¦ cmpt 5226 E cep 5575 Γ cxp 5670 dom cdm 5672 ran crn 5673 βΎ cres 5674 β cima 5675 Oncon0 6364 βcfv 6543 Οcom 7868 reccrdg 8428 βΌ cdom 8960 OrdIsocoi 9532 harchar 9579 TCctc 9759 π 1cr1 9785 rankcrnk 9786 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-smo 8365 df-recs 8390 df-rdg 8429 df-en 8963 df-dom 8964 df-sdom 8965 df-oi 9533 df-har 9580 df-wdom 9588 df-tc 9760 df-r1 9787 df-rank 9788 |
This theorem is referenced by: hsmex 10455 |
Copyright terms: Public domain | W3C validator |