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Mirrors > Home > MPE Home > Th. List > Mathboxes > madef | Structured version Visualization version GIF version |
Description: The made function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.) |
Ref | Expression |
---|---|
madef | β’ M :OnβΆπ« No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-made 34127 | . . 3 β’ M = recs((π₯ β V β¦ ( |s β (π« βͺ ran π₯ Γ π« βͺ ran π₯)))) | |
2 | 1 | tfr1 8298 | . 2 β’ M Fn On |
3 | madeval2 34133 | . . . . . . 7 β’ (π₯ β On β ( M βπ₯) = {π¦ β No β£ βπ§ β π« βͺ ( M β π₯)βπ€ β π« βͺ ( M β π₯)(π§ <<s π€ β§ (π§ |s π€) = π¦)}) | |
4 | ssrab2 4025 | . . . . . . 7 β’ {π¦ β No β£ βπ§ β π« βͺ ( M β π₯)βπ€ β π« βͺ ( M β π₯)(π§ <<s π€ β§ (π§ |s π€) = π¦)} β No | |
5 | 3, 4 | eqsstrdi 3986 | . . . . . 6 β’ (π₯ β On β ( M βπ₯) β No ) |
6 | sseq1 3957 | . . . . . 6 β’ (π¦ = ( M βπ₯) β (π¦ β No β ( M βπ₯) β No )) | |
7 | 5, 6 | syl5ibrcom 246 | . . . . 5 β’ (π₯ β On β (π¦ = ( M βπ₯) β π¦ β No )) |
8 | 7 | rexlimiv 3141 | . . . 4 β’ (βπ₯ β On π¦ = ( M βπ₯) β π¦ β No ) |
9 | vex 3445 | . . . . 5 β’ π¦ β V | |
10 | eqeq1 2740 | . . . . . 6 β’ (π§ = π¦ β (π§ = ( M βπ₯) β π¦ = ( M βπ₯))) | |
11 | 10 | rexbidv 3171 | . . . . 5 β’ (π§ = π¦ β (βπ₯ β On π§ = ( M βπ₯) β βπ₯ β On π¦ = ( M βπ₯))) |
12 | fnrnfv 6885 | . . . . . 6 β’ ( M Fn On β ran M = {π§ β£ βπ₯ β On π§ = ( M βπ₯)}) | |
13 | 2, 12 | ax-mp 5 | . . . . 5 β’ ran M = {π§ β£ βπ₯ β On π§ = ( M βπ₯)} |
14 | 9, 11, 13 | elab2 3623 | . . . 4 β’ (π¦ β ran M β βπ₯ β On π¦ = ( M βπ₯)) |
15 | velpw 4552 | . . . 4 β’ (π¦ β π« No β π¦ β No ) | |
16 | 8, 14, 15 | 3imtr4i 291 | . . 3 β’ (π¦ β ran M β π¦ β π« No ) |
17 | 16 | ssriv 3936 | . 2 β’ ran M β π« No |
18 | df-f 6483 | . 2 β’ ( M :OnβΆπ« No β ( M Fn On β§ ran M β π« No )) | |
19 | 2, 17, 18 | mpbir2an 708 | 1 β’ M :OnβΆπ« No |
Colors of variables: wff setvar class |
Syntax hints: β§ wa 396 = wceq 1540 β wcel 2105 {cab 2713 βwrex 3070 {crab 3403 Vcvv 3441 β wss 3898 π« cpw 4547 βͺ cuni 4852 class class class wbr 5092 β¦ cmpt 5175 Γ cxp 5618 ran crn 5621 β cima 5623 Oncon0 6302 Fn wfn 6474 βΆwf 6475 βcfv 6479 (class class class)co 7337 No csur 26894 <<s csslt 27026 |s cscut 27028 M cmade 34122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-1o 8367 df-2o 8368 df-no 26897 df-slt 26898 df-bday 26899 df-sslt 27027 df-scut 27029 df-made 34127 |
This theorem is referenced by: oldf 34137 newf 34138 madessno 34140 elmade 34147 elold 34149 madess 34155 madeoldsuc 34163 madebdayim 34166 |
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