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Mirrors > Home > ILE Home > Th. List > tfrcllembex | GIF version |
Description: Lemma for tfrcl 6355. The set 𝐵 exists. (Contributed by Jim Kingdon, 25-Mar-2022.) |
Ref | Expression |
---|---|
tfrcl.f | ⊢ 𝐹 = recs(𝐺) |
tfrcl.g | ⊢ (𝜑 → Fun 𝐺) |
tfrcl.x | ⊢ (𝜑 → Ord 𝑋) |
tfrcl.ex | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆) |
tfrcllemsucfn.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} |
tfrcllembacc.3 | ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))} |
tfrcllembacc.u | ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋) |
tfrcllembacc.4 | ⊢ (𝜑 → 𝐷 ∈ 𝑋) |
tfrcllembacc.5 | ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) |
Ref | Expression |
---|---|
tfrcllembex | ⊢ (𝜑 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrcl.f | . . . 4 ⊢ 𝐹 = recs(𝐺) | |
2 | tfrcl.g | . . . 4 ⊢ (𝜑 → Fun 𝐺) | |
3 | tfrcl.x | . . . 4 ⊢ (𝜑 → Ord 𝑋) | |
4 | tfrcl.ex | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆) | |
5 | tfrcllemsucfn.1 | . . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} | |
6 | tfrcllembacc.3 | . . . 4 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))} | |
7 | tfrcllembacc.u | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋) | |
8 | tfrcllembacc.4 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑋) | |
9 | tfrcllembacc.5 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤)))) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | tfrcllembfn 6348 | . . 3 ⊢ (𝜑 → ∪ 𝐵:𝐷⟶𝑆) |
11 | fex 5737 | . . 3 ⊢ ((∪ 𝐵:𝐷⟶𝑆 ∧ 𝐷 ∈ 𝑋) → ∪ 𝐵 ∈ V) | |
12 | 10, 8, 11 | syl2anc 411 | . 2 ⊢ (𝜑 → ∪ 𝐵 ∈ V) |
13 | uniexb 4467 | . 2 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
14 | 12, 13 | sylibr 134 | 1 ⊢ (𝜑 → 𝐵 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∃wex 1490 ∈ wcel 2146 {cab 2161 ∀wral 2453 ∃wrex 2454 Vcvv 2735 ∪ cun 3125 {csn 3589 ⟨cop 3592 ∪ cuni 3805 Ord word 4356 suc csuc 4359 ↾ cres 4622 Fun wfun 5202 ⟶wf 5204 ‘cfv 5208 recscrecs 6295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-ral 2458 df-rex 2459 df-reu 2460 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-iord 4360 df-on 4362 df-suc 4365 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-recs 6296 |
This theorem is referenced by: tfrcllemex 6351 |
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