![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 27901 | . 2 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | |
2 | imassrn 6090 | . . . . . . . 8 ⊢ ( M “ 𝑥) ⊆ ran M | |
3 | madef 27909 | . . . . . . . . 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 4004 | . . . . . . 7 ⊢ ( M “ 𝑥) ⊆ 𝒫 No |
7 | 6 | sseli 3990 | . . . . . 6 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ∈ 𝒫 No ) |
8 | 7 | elpwid 4613 | . . . . 5 ⊢ (𝑦 ∈ ( M “ 𝑥) → 𝑦 ⊆ No ) |
9 | 8 | rgen 3060 | . . . 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 3481 | . . . . . . . 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 7759 | . . . . 5 ⊢ ∪ ( M “ 𝑥) ∈ V |
17 | 16 | elpw 4608 | . . . 4 ⊢ (∪ ( M “ 𝑥) ∈ 𝒫 No ↔ ∪ ( M “ 𝑥) ⊆ No ) |
18 | unissb 4943 | . . . 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 7131 | 1 ⊢ O :On⟶𝒫 No |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 ∀wral 3058 Vcvv 3477 ⊆ wss 3962 𝒫 cpw 4604 ∪ cuni 4911 ran crn 5689 “ cima 5691 Oncon0 6385 Fun wfun 6556 ⟶wf 6558 No csur 27698 M cmade 27895 O cold 27896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-1o 8504 df-2o 8505 df-no 27701 df-slt 27702 df-bday 27703 df-sslt 27840 df-scut 27842 df-made 27900 df-old 27901 |
This theorem is referenced by: oldssno 27914 leftf 27918 rightf 27919 oldssmade 27930 oldlim 27939 oldbdayim 27941 |
Copyright terms: Public domain | W3C validator |