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Mirrors > Home > ILE Home > Th. List > tfr1on | GIF version |
Description: Recursion is defined on an ordinal if the characteristic function is defined up to a suitable point. (Contributed by Jim Kingdon, 12-Mar-2022.) |
Ref | Expression |
---|---|
tfr1on.f | ⊢ 𝐹 = recs(𝐺) |
tfr1on.g | ⊢ (𝜑 → Fun 𝐺) |
tfr1on.x | ⊢ (𝜑 → Ord 𝑋) |
tfr1on.ex | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V) |
tfr1on.u | ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋) |
tfr1on.yx | ⊢ (𝜑 → 𝑌 ∈ 𝑋) |
Ref | Expression |
---|---|
tfr1on | ⊢ (𝜑 → 𝑌 ⊆ dom 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfr1on.f | . 2 ⊢ 𝐹 = recs(𝐺) | |
2 | tfr1on.g | . 2 ⊢ (𝜑 → Fun 𝐺) | |
3 | tfr1on.x | . 2 ⊢ (𝜑 → Ord 𝑋) | |
4 | tfr1on.ex | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V) | |
5 | eqid 2115 | . . 3 ⊢ {𝑎 ∣ ∃𝑏 ∈ 𝑋 (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))} = {𝑎 ∣ ∃𝑏 ∈ 𝑋 (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))} | |
6 | 5 | tfr1onlem3 6189 | . 2 ⊢ {𝑎 ∣ ∃𝑏 ∈ 𝑋 (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))} = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} |
7 | tfr1on.u | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋) | |
8 | tfr1on.yx | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑋) | |
9 | 1, 2, 3, 4, 6, 7, 8 | tfr1onlemres 6200 | 1 ⊢ (𝜑 → 𝑌 ⊆ dom 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 945 = wceq 1314 ∈ wcel 1463 {cab 2101 ∀wral 2390 ∃wrex 2391 Vcvv 2657 ⊆ wss 3037 ∪ cuni 3702 Ord word 4244 suc csuc 4247 dom cdm 4499 ↾ cres 4501 Fun wfun 5075 Fn wfn 5076 ‘cfv 5081 recscrecs 6155 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-tr 3987 df-id 4175 df-iord 4248 df-on 4250 df-suc 4253 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-recs 6156 |
This theorem is referenced by: tfri1dALT 6202 |
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