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Mathbox for Emmett Weisz |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > setrecsres | Structured version Visualization version GIF version |
Description: A recursively generated class is unaffected when its input function is restricted to subsets of the class. (Contributed by Emmett Weisz, 14-Mar-2022.) |
Ref | Expression |
---|---|
setrecsres.1 | ⊢ 𝐵 = setrecs(𝐹) |
setrecsres.2 | ⊢ (𝜑 → Fun 𝐹) |
Ref | Expression |
---|---|
setrecsres | ⊢ (𝜑 → 𝐵 = setrecs((𝐹 ↾ 𝒫 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setrecsres.1 | . . 3 ⊢ 𝐵 = setrecs(𝐹) | |
2 | id 22 | . . . . . . . 8 ⊢ (𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)) → 𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) | |
3 | setrecsres.2 | . . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐹) | |
4 | resss 6001 | . . . . . . . . . . 11 ⊢ (𝐹 ↾ 𝒫 𝐵) ⊆ 𝐹 | |
5 | 4 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ↾ 𝒫 𝐵) ⊆ 𝐹) |
6 | 3, 5 | setrecsss 48243 | . . . . . . . . 9 ⊢ (𝜑 → setrecs((𝐹 ↾ 𝒫 𝐵)) ⊆ setrecs(𝐹)) |
7 | 6, 1 | sseqtrrdi 4024 | . . . . . . . 8 ⊢ (𝜑 → setrecs((𝐹 ↾ 𝒫 𝐵)) ⊆ 𝐵) |
8 | 2, 7 | sylan9ssr 3987 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) → 𝑥 ⊆ 𝐵) |
9 | velpw 4603 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) | |
10 | fvres 6910 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐵 → ((𝐹 ↾ 𝒫 𝐵)‘𝑥) = (𝐹‘𝑥)) | |
11 | 9, 10 | sylbir 234 | . . . . . . 7 ⊢ (𝑥 ⊆ 𝐵 → ((𝐹 ↾ 𝒫 𝐵)‘𝑥) = (𝐹‘𝑥)) |
12 | 8, 11 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) → ((𝐹 ↾ 𝒫 𝐵)‘𝑥) = (𝐹‘𝑥)) |
13 | eqid 2725 | . . . . . . . 8 ⊢ setrecs((𝐹 ↾ 𝒫 𝐵)) = setrecs((𝐹 ↾ 𝒫 𝐵)) | |
14 | vex 3467 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)) → 𝑥 ∈ V) |
16 | 13, 15, 2 | setrec1 48233 | . . . . . . 7 ⊢ (𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)) → ((𝐹 ↾ 𝒫 𝐵)‘𝑥) ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) |
17 | 16 | adantl 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) → ((𝐹 ↾ 𝒫 𝐵)‘𝑥) ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) |
18 | 12, 17 | eqsstrrd 4012 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) → (𝐹‘𝑥) ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) |
19 | 18 | ex 411 | . . . 4 ⊢ (𝜑 → (𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)) → (𝐹‘𝑥) ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)))) |
20 | 19 | alrimiv 1922 | . . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)) → (𝐹‘𝑥) ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)))) |
21 | 1, 20 | setrec2v 48238 | . 2 ⊢ (𝜑 → 𝐵 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) |
22 | 21, 7 | eqssd 3990 | 1 ⊢ (𝜑 → 𝐵 = setrecs((𝐹 ↾ 𝒫 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3463 ⊆ wss 3940 𝒫 cpw 4598 ↾ cres 5674 Fun wfun 6536 ‘cfv 6542 setrecscsetrecs 48225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-reg 9613 ax-inf2 9662 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7418 df-om 7868 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-r1 9785 df-rank 9786 df-setrecs 48226 |
This theorem is referenced by: (None) |
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