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| Mirrors > Home > MPE Home > Th. List > 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 | ⊢ M :On⟶𝒫 No | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-made 27887 | . . 3 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) | |
| 2 | 1 | tfr1 8438 | . 2 ⊢ M Fn On | 
| 3 | madeval2 27893 | . . . . . . 7 ⊢ (𝑥 ∈ On → ( M ‘𝑥) = {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ ( M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)}) | |
| 4 | ssrab2 4079 | . . . . . . 7 ⊢ {𝑦 ∈ No ∣ ∃𝑧 ∈ 𝒫 ∪ ( M “ 𝑥)∃𝑤 ∈ 𝒫 ∪ ( M “ 𝑥)(𝑧 <<s 𝑤 ∧ (𝑧 |s 𝑤) = 𝑦)} ⊆ No | |
| 5 | 3, 4 | eqsstrdi 4027 | . . . . . 6 ⊢ (𝑥 ∈ On → ( M ‘𝑥) ⊆ No ) | 
| 6 | sseq1 4008 | . . . . . 6 ⊢ (𝑦 = ( M ‘𝑥) → (𝑦 ⊆ No ↔ ( M ‘𝑥) ⊆ No )) | |
| 7 | 5, 6 | syl5ibrcom 247 | . . . . 5 ⊢ (𝑥 ∈ On → (𝑦 = ( M ‘𝑥) → 𝑦 ⊆ No )) | 
| 8 | 7 | rexlimiv 3147 | . . . 4 ⊢ (∃𝑥 ∈ On 𝑦 = ( M ‘𝑥) → 𝑦 ⊆ No ) | 
| 9 | vex 3483 | . . . . 5 ⊢ 𝑦 ∈ V | |
| 10 | eqeq1 2740 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 = ( M ‘𝑥) ↔ 𝑦 = ( M ‘𝑥))) | |
| 11 | 10 | rexbidv 3178 | . . . . 5 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ On 𝑧 = ( M ‘𝑥) ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥))) | 
| 12 | fnrnfv 6967 | . . . . . 6 ⊢ ( M Fn On → ran M = {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = ( M ‘𝑥)}) | |
| 13 | 2, 12 | ax-mp 5 | . . . . 5 ⊢ ran M = {𝑧 ∣ ∃𝑥 ∈ On 𝑧 = ( M ‘𝑥)} | 
| 14 | 9, 11, 13 | elab2 3681 | . . . 4 ⊢ (𝑦 ∈ ran M ↔ ∃𝑥 ∈ On 𝑦 = ( M ‘𝑥)) | 
| 15 | velpw 4604 | . . . 4 ⊢ (𝑦 ∈ 𝒫 No ↔ 𝑦 ⊆ No ) | |
| 16 | 8, 14, 15 | 3imtr4i 292 | . . 3 ⊢ (𝑦 ∈ ran M → 𝑦 ∈ 𝒫 No ) | 
| 17 | 16 | ssriv 3986 | . 2 ⊢ ran M ⊆ 𝒫 No | 
| 18 | df-f 6564 | . 2 ⊢ ( M :On⟶𝒫 No ↔ ( M Fn On ∧ ran M ⊆ 𝒫 No )) | |
| 19 | 2, 17, 18 | mpbir2an 711 | 1 ⊢ M :On⟶𝒫 No | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1539 ∈ wcel 2107 {cab 2713 ∃wrex 3069 {crab 3435 Vcvv 3479 ⊆ wss 3950 𝒫 cpw 4599 ∪ cuni 4906 class class class wbr 5142 ↦ cmpt 5224 × cxp 5682 ran crn 5685 “ cima 5687 Oncon0 6383 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 No csur 27685 <<s csslt 27826 |s cscut 27828 M cmade 27882 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-1o 8507 df-2o 8508 df-no 27688 df-slt 27689 df-bday 27690 df-sslt 27827 df-scut 27829 df-made 27887 | 
| This theorem is referenced by: oldf 27897 newf 27898 madessno 27900 elmade 27907 elold 27909 old1 27915 madess 27916 madeoldsuc 27924 madebdayim 27927 madefi 27951 oldfi 27952 | 
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