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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 34031 | . . . . 5 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) | |
2 | recsfnon 8234 | . . . . . . 7 ⊢ recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On | |
3 | fnfun 6533 | . . . . . . 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 6454 | . . . . . 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 6520 | . . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V) | |
9 | 7, 8 | mpan 687 | . . 3 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V) |
10 | 9 | uniexd 7595 | . 2 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) ∈ V) |
11 | imaeq2 5965 | . . . 4 ⊢ (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴)) | |
12 | 11 | unieqd 4853 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ( M “ 𝑥) = ∪ ( M “ 𝐴)) |
13 | df-old 34032 | . . 3 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | |
14 | 12, 13 | fvmptg 6873 | . 2 ⊢ ((𝐴 ∈ On ∧ ∪ ( M “ 𝐴) ∈ V) → ( O ‘𝐴) = ∪ ( M “ 𝐴)) |
15 | 10, 14 | mpdan 684 | 1 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 Vcvv 3432 𝒫 cpw 4533 ∪ cuni 4839 ↦ cmpt 5157 × cxp 5587 ran crn 5590 “ cima 5592 Oncon0 6266 Fun wfun 6427 Fn wfn 6428 ‘cfv 6433 recscrecs 8201 |s cscut 33977 M cmade 34026 O cold 34027 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-made 34031 df-old 34032 |
This theorem is referenced by: old0 34043 elmade2 34052 elold 34053 madeoldsuc 34067 |
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