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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 34060 | . 2 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | |
2 | imassrn 5981 | . . . . . . . 8 ⊢ ( M “ 𝑥) ⊆ ran M | |
3 | madef 34068 | . . . . . . . . 9 ⊢ M :On⟶𝒫 No | |
4 | frn 6625 | . . . . . . . . 9 ⊢ ( M :On⟶𝒫 No → ran M ⊆ 𝒫 No ) | |
5 | 3, 4 | ax-mp 5 | . . . . . . . 8 ⊢ ran M ⊆ 𝒫 No |
6 | 2, 5 | sstri 3932 | . . . . . . 7 ⊢ ( M “ 𝑥) ⊆ 𝒫 No |
7 | 6 | sseli 3919 | . . . . . 6 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ∈ 𝒫 No ) |
8 | 7 | elpwid 4547 | . . . . 5 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ⊆ No ) |
9 | 8 | rgen 3061 | . . . 4 ⊢ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No |
10 | 9 | a1i 11 | . . 3 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No ) |
11 | ffun 6621 | . . . . . . . 8 ⊢ ( M :On⟶𝒫 No → Fun M ) | |
12 | 3, 11 | ax-mp 5 | . . . . . . 7 ⊢ Fun M |
13 | vex 3438 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
14 | 13 | funimaex 6539 | . . . . . . 7 ⊢ (Fun M → ( M “ 𝑥) ∈ V) |
15 | 12, 14 | ax-mp 5 | . . . . . 6 ⊢ ( M “ 𝑥) ∈ V |
16 | 15 | uniex 7614 | . . . . 5 ⊢ ∪ ( M “ 𝑥) ∈ V |
17 | 16 | elpw 4540 | . . . 4 ⊢ (∪ ( M “ 𝑥) ∈ 𝒫 No ↔ ∪ ( M “ 𝑥) ⊆ No ) |
18 | unissb 4876 | . . . 4 ⊢ (∪ ( M “ 𝑥) ⊆ No ↔ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No ) | |
19 | 17, 18 | bitri 274 | . . 3 ⊢ (∪ ( M “ 𝑥) ∈ 𝒫 No ↔ ∀𝑦 ∈ ( M “ 𝑥)𝑦 ⊆ No ) |
20 | 10, 19 | sylibr 233 | . 2 ⊢ (𝑥 ∈ On → ∪ ( M “ 𝑥) ∈ 𝒫 No ) |
21 | 1, 20 | fmpti 7006 | 1 ⊢ O :On⟶𝒫 No |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2101 ∀wral 3059 Vcvv 3434 ⊆ wss 3889 𝒫 cpw 4536 ∪ cuni 4841 ran crn 5592 “ cima 5594 Oncon0 6270 Fun wfun 6441 ⟶wf 6443 No csur 33871 M cmade 34054 O cold 34055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3222 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3908 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4842 df-int 4883 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-tr 5195 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6206 df-ord 6273 df-on 6274 df-suc 6276 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-riota 7252 df-ov 7298 df-oprab 7299 df-mpo 7300 df-2nd 7852 df-frecs 8117 df-wrecs 8148 df-recs 8222 df-1o 8317 df-2o 8318 df-no 33874 df-slt 33875 df-bday 33876 df-sslt 34004 df-scut 34006 df-made 34059 df-old 34060 |
This theorem is referenced by: oldssno 34073 leftf 34077 rightf 34078 oldssmade 34088 oldlim 34097 oldbdayim 34099 |
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