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| Mirrors > Home > MPE Home > Th. List > oldf | Structured version Visualization version GIF version | ||
| Description: The older function is a function from ordinals to sets of surreals. (Contributed by Scott Fenton, 6-Aug-2024.) |
| Ref | Expression |
|---|---|
| oldf | ⊢ O :On⟶𝒫 No |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-old 27887 | . 2 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | |
| 2 | imassrn 6089 | . . . . . . . 8 ⊢ ( M “ 𝑥) ⊆ ran M | |
| 3 | madef 27895 | . . . . . . . . 9 ⊢ M :On⟶𝒫 No | |
| 4 | frn 6743 | . . . . . . . . 9 ⊢ ( M :On⟶𝒫 No → ran M ⊆ 𝒫 No ) | |
| 5 | 3, 4 | ax-mp 5 | . . . . . . . 8 ⊢ ran M ⊆ 𝒫 No |
| 6 | 2, 5 | sstri 3993 | . . . . . . 7 ⊢ ( M “ 𝑥) ⊆ 𝒫 No |
| 7 | 6 | sseli 3979 | . . . . . 6 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ∈ 𝒫 No ) |
| 8 | 7 | elpwid 4609 | . . . . 5 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ⊆ No ) |
| 9 | 8 | rgen 3063 | . . . 4 ⊢ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No |
| 10 | 9 | a1i 11 | . . 3 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No ) |
| 11 | ffun 6739 | . . . . . . . 8 ⊢ ( M :On⟶𝒫 No → Fun M ) | |
| 12 | 3, 11 | ax-mp 5 | . . . . . . 7 ⊢ Fun M |
| 13 | vex 3484 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 14 | 13 | funimaex 6655 | . . . . . . 7 ⊢ (Fun M → ( M “ 𝑥) ∈ V) |
| 15 | 12, 14 | ax-mp 5 | . . . . . 6 ⊢ ( M “ 𝑥) ∈ V |
| 16 | 15 | uniex 7761 | . . . . 5 ⊢ ∪ ( M “ 𝑥) ∈ V |
| 17 | 16 | elpw 4604 | . . . 4 ⊢ (∪ ( M “ 𝑥) ∈ 𝒫 No ↔ ∪ ( M “ 𝑥) ⊆ No ) |
| 18 | unissb 4939 | . . . 4 ⊢ (∪ ( M “ 𝑥) ⊆ No ↔ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No ) | |
| 19 | 17, 18 | bitri 275 | . . 3 ⊢ (∪ ( M “ 𝑥) ∈ 𝒫 No ↔ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No ) |
| 20 | 10, 19 | sylibr 234 | . 2 ⊢ (𝑥 ∈ On → ∪ ( M “ 𝑥) ∈ 𝒫 No ) |
| 21 | 1, 20 | fmpti 7132 | 1 ⊢ O :On⟶𝒫 No |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 ran crn 5686 “ cima 5688 Oncon0 6384 Fun wfun 6555 ⟶wf 6557 No csur 27684 M cmade 27881 O cold 27882 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-1o 8506 df-2o 8507 df-no 27687 df-slt 27688 df-bday 27689 df-sslt 27826 df-scut 27828 df-made 27886 df-old 27887 |
| This theorem is referenced by: oldssno 27900 leftf 27904 rightf 27905 oldssmade 27916 oldlim 27925 oldbdayim 27927 |
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