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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 33626 | . 2 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | |
2 | imassrn 5917 | . . . . . . . 8 ⊢ ( M “ 𝑥) ⊆ ran M | |
3 | madef 33634 | . . . . . . . . 9 ⊢ M :On⟶𝒫 No | |
4 | frn 6509 | . . . . . . . . 9 ⊢ ( M :On⟶𝒫 No → ran M ⊆ 𝒫 No ) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . 8 ⊢ ran M ⊆ 𝒫 No |
6 | 2, 5 | sstri 3903 | . . . . . . 7 ⊢ ( M “ 𝑥) ⊆ 𝒫 No |
7 | 6 | sseli 3890 | . . . . . 6 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ∈ 𝒫 No ) |
8 | 7 | elpwid 4508 | . . . . 5 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ⊆ No ) |
9 | 8 | rgen 3080 | . . . 4 ⊢ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No ) |
11 | ffun 6506 | . . . . . . . 8 ⊢ ( M :On⟶𝒫 No → Fun M ) | |
12 | 3, 11 | ax-mp 5 | . . . . . . 7 ⊢ Fun M |
13 | vex 3413 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
14 | 13 | funimaex 6427 | . . . . . . 7 ⊢ (Fun M → ( M “ 𝑥) ∈ V) |
15 | 12, 14 | ax-mp 5 | . . . . . 6 ⊢ ( M “ 𝑥) ∈ V |
16 | 15 | uniex 7471 | . . . . 5 ⊢ ∪ ( M “ 𝑥) ∈ V |
17 | 16 | elpw 4501 | . . . 4 ⊢ (∪ ( M “ 𝑥) ∈ 𝒫 No ↔ ∪ ( M “ 𝑥) ⊆ No ) |
18 | unissb 4835 | . . . 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 6873 | 1 ⊢ O :On⟶𝒫 No |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ∀wral 3070 Vcvv 3409 ⊆ wss 3860 𝒫 cpw 4497 ∪ cuni 4801 ran crn 5529 “ cima 5531 Oncon0 6174 Fun wfun 6334 ⟶wf 6336 No csur 33440 M cmade 33620 O cold 33621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-wrecs 7963 df-recs 8024 df-1o 8118 df-2o 8119 df-no 33443 df-slt 33444 df-bday 33445 df-sslt 33573 df-scut 33575 df-made 33625 df-old 33626 |
This theorem is referenced by: leftf 33639 rightf 33640 leftssno 33654 rightssno 33655 oldlim 33660 lrold 33668 lrrecpred 33683 |
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