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Mirrors > Home > MPE Home > Th. List > hsmexlem6 | Structured version Visualization version GIF version |
Description: Lemma for hsmex 10429. (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 6898 | . 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 10427 | . . . 4 β’ (π β π β (rankβπ) β (harβπ« (Ο Γ βͺ ran π»))) |
8 | 5 | ssrab3 4075 | . . . . . 6 β’ π β βͺ (π 1 β On) |
9 | 8 | sseli 3973 | . . . . 5 β’ (π β π β π β βͺ (π 1 β On)) |
10 | harcl 9556 | . . . . . 6 β’ (harβπ« (Ο Γ βͺ ran π»)) β On | |
11 | r1fnon 9764 | . . . . . . 7 β’ π 1 Fn On | |
12 | 11 | fndmi 6647 | . . . . . 6 β’ dom π 1 = On |
13 | 10, 12 | eleqtrri 2826 | . . . . 5 β’ (harβπ« (Ο Γ βͺ ran π»)) β dom π 1 |
14 | rankr1ag 9799 | . . . . 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 585 | . . . 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 3981 | . 2 β’ π β (π 1β(harβπ« (Ο Γ βͺ ran π»))) |
18 | 1, 17 | ssexi 5315 | 1 β’ π β V |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1533 β wcel 2098 βwral 3055 {crab 3426 Vcvv 3468 π« cpw 4597 {csn 4623 βͺ cuni 4902 class class class wbr 5141 β¦ cmpt 5224 E cep 5572 Γ cxp 5667 dom cdm 5669 ran crn 5670 βΎ cres 5671 β cima 5672 Oncon0 6358 βcfv 6537 Οcom 7852 reccrdg 8410 βΌ cdom 8939 OrdIsocoi 9506 harchar 9553 TCctc 9733 π 1cr1 9759 rankcrnk 9760 |
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 ax-inf2 9638 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 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-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 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-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 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-isom 6546 df-riota 7361 df-ov 7408 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-smo 8347 df-recs 8372 df-rdg 8411 df-en 8942 df-dom 8943 df-sdom 8944 df-oi 9507 df-har 9554 df-wdom 9562 df-tc 9734 df-r1 9761 df-rank 9762 |
This theorem is referenced by: hsmex 10429 |
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