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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 33672 | . . . . 5 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) | |
2 | recsfnon 8068 | . . . . . . 7 ⊢ recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On | |
3 | fnfun 6438 | . . . . . . 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 6359 | . . . . . 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 261 | . . . . 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 6425 | . . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V) | |
9 | 7, 8 | mpan 690 | . . 3 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V) |
10 | 9 | uniexd 7486 | . 2 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) ∈ V) |
11 | imaeq2 5899 | . . . 4 ⊢ (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴)) | |
12 | 11 | unieqd 4810 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ( M “ 𝑥) = ∪ ( M “ 𝐴)) |
13 | df-old 33673 | . . 3 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | |
14 | 12, 13 | fvmptg 6773 | . 2 ⊢ ((𝐴 ∈ On ∧ ∪ ( M “ 𝐴) ∈ V) → ( O ‘𝐴) = ∪ ( M “ 𝐴)) |
15 | 10, 14 | mpdan 687 | 1 ⊢ (𝐴 ∈ On → ( O ‘𝐴) = ∪ ( M “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 Vcvv 3398 𝒫 cpw 4488 ∪ cuni 4796 ↦ cmpt 5110 × cxp 5523 ran crn 5526 “ cima 5528 Oncon0 6172 Fun wfun 6333 Fn wfn 6334 ‘cfv 6339 recscrecs 8036 |s cscut 33618 M cmade 33667 O cold 33668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-wrecs 7976 df-recs 8037 df-made 33672 df-old 33673 |
This theorem is referenced by: elmade2 33689 elold 33690 old0 33694 madeoldsuc 33705 |
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