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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 33795 | . 2 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | |
2 | imassrn 5955 | . . . . . . . 8 ⊢ ( M “ 𝑥) ⊆ ran M | |
3 | madef 33803 | . . . . . . . . 9 ⊢ M :On⟶𝒫 No | |
4 | frn 6571 | . . . . . . . . 9 ⊢ ( M :On⟶𝒫 No → ran M ⊆ 𝒫 No ) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . 8 ⊢ ran M ⊆ 𝒫 No |
6 | 2, 5 | sstri 3925 | . . . . . . 7 ⊢ ( M “ 𝑥) ⊆ 𝒫 No |
7 | 6 | sseli 3911 | . . . . . 6 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ∈ 𝒫 No ) |
8 | 7 | elpwid 4539 | . . . . 5 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ⊆ No ) |
9 | 8 | rgen 3072 | . . . 4 ⊢ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No ) |
11 | ffun 6567 | . . . . . . . 8 ⊢ ( M :On⟶𝒫 No → Fun M ) | |
12 | 3, 11 | ax-mp 5 | . . . . . . 7 ⊢ Fun M |
13 | vex 3425 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
14 | 13 | funimaex 6485 | . . . . . . 7 ⊢ (Fun M → ( M “ 𝑥) ∈ V) |
15 | 12, 14 | ax-mp 5 | . . . . . 6 ⊢ ( M “ 𝑥) ∈ V |
16 | 15 | uniex 7548 | . . . . 5 ⊢ ∪ ( M “ 𝑥) ∈ V |
17 | 16 | elpw 4532 | . . . 4 ⊢ (∪ ( M “ 𝑥) ∈ 𝒫 No ↔ ∪ ( M “ 𝑥) ⊆ No ) |
18 | unissb 4868 | . . . 4 ⊢ (∪ ( M “ 𝑥) ⊆ No ↔ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No ) | |
19 | 17, 18 | bitri 278 | . . 3 ⊢ (∪ ( M “ 𝑥) ∈ 𝒫 No ↔ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No ) |
20 | 10, 19 | sylibr 237 | . 2 ⊢ (𝑥 ∈ On → ∪ ( M “ 𝑥) ∈ 𝒫 No ) |
21 | 1, 20 | fmpti 6948 | 1 ⊢ O :On⟶𝒫 No |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ∀wral 3062 Vcvv 3421 ⊆ wss 3881 𝒫 cpw 4528 ∪ cuni 4834 ran crn 5567 “ cima 5569 Oncon0 6231 Fun wfun 6392 ⟶wf 6394 No csur 33606 M cmade 33789 O cold 33790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-int 4875 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-wrecs 8068 df-recs 8129 df-1o 8223 df-2o 8224 df-no 33609 df-slt 33610 df-bday 33611 df-sslt 33739 df-scut 33741 df-made 33794 df-old 33795 |
This theorem is referenced by: oldssno 33808 leftf 33812 rightf 33813 oldssmade 33823 oldlim 33832 oldbdayim 33834 |
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