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Mirrors > Home > MPE Home > Th. List > ulmdvlem2 | Structured version Visualization version GIF version |
Description: Lemma for ulmdv 25778. (Contributed by Mario Carneiro, 8-May-2015.) |
Ref | Expression |
---|---|
ulmdv.z | β’ π = (β€β₯βπ) |
ulmdv.s | β’ (π β π β {β, β}) |
ulmdv.m | β’ (π β π β β€) |
ulmdv.f | β’ (π β πΉ:πβΆ(β βm π)) |
ulmdv.g | β’ (π β πΊ:πβΆβ) |
ulmdv.l | β’ ((π β§ π§ β π) β (π β π β¦ ((πΉβπ)βπ§)) β (πΊβπ§)) |
ulmdv.u | β’ (π β (π β π β¦ (π D (πΉβπ)))(βπ’βπ)π») |
Ref | Expression |
---|---|
ulmdvlem2 | β’ ((π β§ π β π) β dom (π D (πΉβπ)) = π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7395 | . . . . . 6 β’ (π D (πΉβπ)) β V | |
2 | 1 | rgenw 3069 | . . . . 5 β’ βπ β π (π D (πΉβπ)) β V |
3 | eqid 2737 | . . . . . 6 β’ (π β π β¦ (π D (πΉβπ))) = (π β π β¦ (π D (πΉβπ))) | |
4 | 3 | fnmpt 6646 | . . . . 5 β’ (βπ β π (π D (πΉβπ)) β V β (π β π β¦ (π D (πΉβπ))) Fn π) |
5 | 2, 4 | mp1i 13 | . . . 4 β’ (π β (π β π β¦ (π D (πΉβπ))) Fn π) |
6 | ulmdv.u | . . . 4 β’ (π β (π β π β¦ (π D (πΉβπ)))(βπ’βπ)π») | |
7 | ulmf2 25759 | . . . 4 β’ (((π β π β¦ (π D (πΉβπ))) Fn π β§ (π β π β¦ (π D (πΉβπ)))(βπ’βπ)π») β (π β π β¦ (π D (πΉβπ))):πβΆ(β βm π)) | |
8 | 5, 6, 7 | syl2anc 585 | . . 3 β’ (π β (π β π β¦ (π D (πΉβπ))):πβΆ(β βm π)) |
9 | 8 | fvmptelcdm 7066 | . 2 β’ ((π β§ π β π) β (π D (πΉβπ)) β (β βm π)) |
10 | elmapi 8794 | . 2 β’ ((π D (πΉβπ)) β (β βm π) β (π D (πΉβπ)):πβΆβ) | |
11 | fdm 6682 | . 2 β’ ((π D (πΉβπ)):πβΆβ β dom (π D (πΉβπ)) = π) | |
12 | 9, 10, 11 | 3syl 18 | 1 β’ ((π β§ π β π) β dom (π D (πΉβπ)) = π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 Vcvv 3448 {cpr 4593 class class class wbr 5110 β¦ cmpt 5193 dom cdm 5638 Fn wfn 6496 βΆwf 6497 βcfv 6501 (class class class)co 7362 βm cmap 8772 βcc 11056 βcr 11057 β€cz 12506 β€β₯cuz 12770 β cli 15373 D cdv 25243 βπ’culm 25751 |
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 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 |
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-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-map 8774 df-pm 8775 df-neg 11395 df-z 12507 df-uz 12771 df-ulm 25752 |
This theorem is referenced by: ulmdvlem3 25777 ulmdv 25778 |
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