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Mirrors > Home > ILE Home > Th. List > freccl | GIF version |
Description: Closure for finite recursion. (Contributed by Jim Kingdon, 27-Mar-2022.) |
Ref | Expression |
---|---|
freccl.a | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
freccl.cl | ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆) |
freccl.b | ⊢ (𝜑 → 𝐵 ∈ ω) |
Ref | Expression |
---|---|
freccl | ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | freccl.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
2 | freccl.cl | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆) | |
3 | freccl.b | . 2 ⊢ (𝜑 → 𝐵 ∈ ω) | |
4 | eqid 2140 | . 2 ⊢ recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) | |
5 | 1, 2, 3, 4 | freccllem 6307 | 1 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ wo 698 = wceq 1332 ∈ wcel 1481 {cab 2126 ∃wrex 2418 Vcvv 2689 ∅c0 3368 ↦ cmpt 3997 suc csuc 4295 ωcom 4512 dom cdm 4547 ‘cfv 5131 recscrecs 6209 freccfrec 6295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-recs 6210 df-frec 6296 |
This theorem is referenced by: frec2uzzd 10204 frecuzrdgrrn 10212 |
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