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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 5963 | . . . . . . . . . . 11 ⊢ (𝐹 ↾ 𝒫 𝐵) ⊆ 𝐹 | |
5 | 4 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ↾ 𝒫 𝐵) ⊆ 𝐹) |
6 | 3, 5 | setrecsss 47232 | . . . . . . . . 9 ⊢ (𝜑 → setrecs((𝐹 ↾ 𝒫 𝐵)) ⊆ setrecs(𝐹)) |
7 | 6, 1 | sseqtrrdi 3996 | . . . . . . . 8 ⊢ (𝜑 → setrecs((𝐹 ↾ 𝒫 𝐵)) ⊆ 𝐵) |
8 | 2, 7 | sylan9ssr 3959 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) → 𝑥 ⊆ 𝐵) |
9 | velpw 4566 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵) | |
10 | fvres 6862 | . . . . . . . 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 2733 | . . . . . . . 8 ⊢ setrecs((𝐹 ↾ 𝒫 𝐵)) = setrecs((𝐹 ↾ 𝒫 𝐵)) | |
14 | vex 3448 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
15 | 14 | a1i 11 | . . . . . . . 8 ⊢ (𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)) → 𝑥 ∈ V) |
16 | 13, 15, 2 | setrec1 47222 | . . . . . . 7 ⊢ (𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)) → ((𝐹 ↾ 𝒫 𝐵)‘𝑥) ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) |
17 | 16 | adantl 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) → ((𝐹 ↾ 𝒫 𝐵)‘𝑥) ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) |
18 | 12, 17 | eqsstrrd 3984 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) → (𝐹‘𝑥) ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) |
19 | 18 | ex 414 | . . . 4 ⊢ (𝜑 → (𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)) → (𝐹‘𝑥) ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)))) |
20 | 19 | alrimiv 1931 | . . 3 ⊢ (𝜑 → ∀𝑥(𝑥 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)) → (𝐹‘𝑥) ⊆ setrecs((𝐹 ↾ 𝒫 𝐵)))) |
21 | 1, 20 | setrec2v 47227 | . 2 ⊢ (𝜑 → 𝐵 ⊆ setrecs((𝐹 ↾ 𝒫 𝐵))) |
22 | 21, 7 | eqssd 3962 | 1 ⊢ (𝜑 → 𝐵 = setrecs((𝐹 ↾ 𝒫 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 𝒫 cpw 4561 ↾ cres 5636 Fun wfun 6491 ‘cfv 6497 setrecscsetrecs 47214 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-reg 9533 ax-inf2 9582 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 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-int 4909 df-iun 4957 df-iin 4958 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-lim 6323 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-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-r1 9705 df-rank 9706 df-setrecs 47215 |
This theorem is referenced by: (None) |
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