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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 27180 | . . . . 5 ⊢ M = recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) | |
2 | recsfnon 8350 | . . . . . . 7 ⊢ recs((𝑥 ∈ V ↦ ( |s “ (𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥)))) Fn On | |
3 | fnfun 6603 | . . . . . . 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 6522 | . . . . . 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 6588 | . . . 4 ⊢ ((Fun M ∧ 𝐴 ∈ On) → ( M “ 𝐴) ∈ V) | |
9 | 7, 8 | mpan 689 | . . 3 ⊢ (𝐴 ∈ On → ( M “ 𝐴) ∈ V) |
10 | 9 | uniexd 7680 | . 2 ⊢ (𝐴 ∈ On → ∪ ( M “ 𝐴) ∈ V) |
11 | imaeq2 6010 | . . . 4 ⊢ (𝑥 = 𝐴 → ( M “ 𝑥) = ( M “ 𝐴)) | |
12 | 11 | unieqd 4880 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ ( M “ 𝑥) = ∪ ( M “ 𝐴)) |
13 | df-old 27181 | . . 3 ⊢ O = (𝑥 ∈ On ↦ ∪ ( M “ 𝑥)) | |
14 | 12, 13 | fvmptg 6947 | . 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 3446 𝒫 cpw 4561 ∪ cuni 4866 ↦ cmpt 5189 × cxp 5632 ran crn 5635 “ cima 5637 Oncon0 6318 Fun wfun 6491 Fn wfn 6492 ‘cfv 6497 recscrecs 8317 |s cscut 27125 M cmade 27175 O cold 27176 |
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 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-made 27180 df-old 27181 |
This theorem is referenced by: old0 27192 elmade2 27201 elold 27202 old1 27208 madeoldsuc 27217 |
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