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Mirrors > Home > ILE Home > Th. List > tfr1onlembex | GIF version |
Description: Lemma for tfr1on 6115. The set 𝐵 exists. (Contributed by Jim Kingdon, 14-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) |
tfr1onlemsucfn.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} |
tfr1onlembacc.3 | ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}))} |
tfr1onlembacc.u | ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋) |
tfr1onlembacc.4 | ⊢ (𝜑 → 𝐷 ∈ 𝑋) |
tfr1onlembacc.5 | ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) |
Ref | Expression |
---|---|
tfr1onlembex | ⊢ (𝜑 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfr1on.f | . . . 4 ⊢ 𝐹 = recs(𝐺) | |
2 | tfr1on.g | . . . 4 ⊢ (𝜑 → Fun 𝐺) | |
3 | tfr1on.x | . . . 4 ⊢ (𝜑 → Ord 𝑋) | |
4 | tfr1on.ex | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V) | |
5 | tfr1onlemsucfn.1 | . . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} | |
6 | tfr1onlembacc.3 | . . . 4 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}))} | |
7 | tfr1onlembacc.u | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋) | |
8 | tfr1onlembacc.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑋) | |
9 | tfr1onlembacc.5 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | tfr1onlembfn 6109 | . . 3 ⊢ (𝜑 → ∪ 𝐵 Fn 𝐷) |
11 | fnex 5519 | . . 3 ⊢ ((∪ 𝐵 Fn 𝐷 ∧ 𝐷 ∈ 𝑋) → ∪ 𝐵 ∈ V) | |
12 | 10, 8, 11 | syl2anc 403 | . 2 ⊢ (𝜑 → ∪ 𝐵 ∈ V) |
13 | uniexb 4295 | . 2 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
14 | 12, 13 | sylibr 132 | 1 ⊢ (𝜑 → 𝐵 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 924 = wceq 1289 ∃wex 1426 ∈ wcel 1438 {cab 2074 ∀wral 2359 ∃wrex 2360 Vcvv 2619 ∪ cun 2997 {csn 3446 〈cop 3449 ∪ cuni 3653 Ord word 4189 suc csuc 4192 ↾ cres 4440 Fun wfun 5009 Fn wfn 5010 ‘cfv 5015 recscrecs 6069 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-id 4120 df-iord 4193 df-on 4195 df-suc 4198 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-recs 6070 |
This theorem is referenced by: tfr1onlemex 6112 |
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