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Mirrors > Home > ILE Home > Th. List > tfrcllemex | GIF version |
Description: Lemma for tfrcl 6269. (Contributed by Jim Kingdon, 26-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 |
---|---|
tfrcllemex | ⊢ (𝜑 → ∃𝑓(𝑓:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)))) |
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 | tfrcllembex 6263 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | uniexg 4369 | . . 3 ⊢ (𝐵 ∈ V → ∪ 𝐵 ∈ V) | |
12 | 10, 11 | syl 14 | . 2 ⊢ (𝜑 → ∪ 𝐵 ∈ V) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | tfrcllembfn 6262 | . . 3 ⊢ (𝜑 → ∪ 𝐵:𝐷⟶𝑆) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | tfrcllemubacc 6264 | . . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))) |
15 | 13, 14 | jca 304 | . 2 ⊢ (𝜑 → (∪ 𝐵:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))) |
16 | feq1 5263 | . . . 4 ⊢ (𝑓 = ∪ 𝐵 → (𝑓:𝐷⟶𝑆 ↔ ∪ 𝐵:𝐷⟶𝑆)) | |
17 | fveq1 5428 | . . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝑓‘𝑢) = (∪ 𝐵‘𝑢)) | |
18 | reseq1 4821 | . . . . . . 7 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑢)) | |
19 | 18 | fveq2d 5433 | . . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝐺‘(𝑓 ↾ 𝑢)) = (𝐺‘(∪ 𝐵 ↾ 𝑢))) |
20 | 17, 19 | eqeq12d 2155 | . . . . 5 ⊢ (𝑓 = ∪ 𝐵 → ((𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))) |
21 | 20 | ralbidv 2438 | . . . 4 ⊢ (𝑓 = ∪ 𝐵 → (∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))) |
22 | 16, 21 | anbi12d 465 | . . 3 ⊢ (𝑓 = ∪ 𝐵 → ((𝑓:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢))) ↔ (∪ 𝐵:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))))) |
23 | 22 | spcegv 2777 | . 2 ⊢ (∪ 𝐵 ∈ V → ((∪ 𝐵:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))) → ∃𝑓(𝑓:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢))))) |
24 | 12, 15, 23 | sylc 62 | 1 ⊢ (𝜑 → ∃𝑓(𝑓:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 = wceq 1332 ∃wex 1469 ∈ wcel 1481 {cab 2126 ∀wral 2417 ∃wrex 2418 Vcvv 2689 ∪ cun 3074 {csn 3532 〈cop 3535 ∪ cuni 3744 Ord word 4292 suc csuc 4295 ↾ cres 4549 Fun wfun 5125 ⟶wf 5127 ‘cfv 5131 recscrecs 6209 |
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-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 |
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-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 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 |
This theorem is referenced by: tfrcllemaccex 6266 |
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