![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > oldval | Structured version Visualization version GIF version |
Description: The value of the old options function. (Contributed by Scott Fenton, 6-Aug-2024.) |
Ref | Expression |
---|---|
oldval | ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-made 27332 | . . . . 5 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) | |
2 | recsfnon 8400 | . . . . . . 7 ⊢ recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On | |
3 | fnfun 6647 | . . . . . . 7 ⊢ (recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On → Fun recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥))))) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ Fun recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) |
5 | funeq 6566 | . . . . . 6 ⊢ ( M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) → (Fun M ↔ Fun recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))))) | |
6 | 4, 5 | mpbiri 258 | . . . . 5 ⊢ ( M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) → Fun M ) |
7 | 1, 6 | ax-mp 5 | . . . 4 ⊢ Fun M |
8 | funimaexg 6632 | . . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V) | |
9 | 7, 8 | mpan 689 | . . 3 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V) |
10 | 9 | uniexd 7729 | . 2 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) ∈ V) |
11 | imaeq2 6054 | . . . 4 ⊢ (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴)) | |
12 | 11 | unieqd 4922 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ( M “ 𝑥) = ∪ ( M “ 𝐴)) |
13 | df-old 27333 | . . 3 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | |
14 | 12, 13 | fvmptg 6994 | . 2 ⊢ ((𝐴 ∈ On ∧ ∪ ( M “ 𝐴) ∈ V) → ( O ‘𝐴) = ∪ ( M “ 𝐴)) |
15 | 10, 14 | mpdan 686 | 1 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3475 𝒫 cpw 4602 ∪ cuni 4908 ↦ cmpt 5231 × cxp 5674 ran crn 5677 “ cima 5679 Oncon0 6362 Fun wfun 6535 Fn wfn 6536 ‘cfv 6541 recscrecs 8367 |s cscut 27274 M cmade 27327 O cold 27328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-made 27332 df-old 27333 |
This theorem is referenced by: old0 27344 elmade2 27353 elold 27354 old1 27360 madeoldsuc 27369 |
Copyright terms: Public domain | W3C validator |