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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 27123 | . . . . 5 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) | |
2 | recsfnon 8341 | . . . . . . 7 ⊢ recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On | |
3 | fnfun 6599 | . . . . . . 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 6518 | . . . . . 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 257 | . . . . 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 6584 | . . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V) | |
9 | 7, 8 | mpan 688 | . . 3 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V) |
10 | 9 | uniexd 7671 | . 2 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) ∈ V) |
11 | imaeq2 6007 | . . . 4 ⊢ (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴)) | |
12 | 11 | unieqd 4877 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ( M “ 𝑥) = ∪ ( M “ 𝐴)) |
13 | df-old 27124 | . . 3 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | |
14 | 12, 13 | fvmptg 6943 | . 2 ⊢ ((𝐴 ∈ On ∧ ∪ ( M “ 𝐴) ∈ V) → ( O ‘𝐴) = ∪ ( M “ 𝐴)) |
15 | 10, 14 | mpdan 685 | 1 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3443 𝒫 cpw 4558 ∪ cuni 4863 ↦ cmpt 5186 × cxp 5629 ran crn 5632 “ cima 5634 Oncon0 6315 Fun wfun 6487 Fn wfn 6488 ‘cfv 6493 recscrecs 8308 |s cscut 27068 M cmade 27118 O cold 27119 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-made 27123 df-old 27124 |
This theorem is referenced by: old0 27135 elmade2 27144 elold 27145 madeoldsuc 27159 |
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