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Mirrors > Home > ILE Home > Th. List > tfr1onlemex | GIF version |
Description: Lemma for tfr1on 6129. (Contributed by Jim Kingdon, 16-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 |
---|---|
tfr1onlemex | ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)))) |
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 | tfr1onlembex 6124 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | uniexg 4275 | . . 3 ⊢ (𝐵 ∈ V → ∪ 𝐵 ∈ V) | |
12 | 10, 11 | syl 14 | . 2 ⊢ (𝜑 → ∪ 𝐵 ∈ V) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | tfr1onlembfn 6123 | . . 3 ⊢ (𝜑 → ∪ 𝐵 Fn 𝐷) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | tfr1onlemubacc 6125 | . . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))) |
15 | 13, 14 | jca 301 | . 2 ⊢ (𝜑 → (∪ 𝐵 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))) |
16 | fneq1 5115 | . . . 4 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 Fn 𝐷 ↔ ∪ 𝐵 Fn 𝐷)) | |
17 | fveq1 5317 | . . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝑓‘𝑢) = (∪ 𝐵‘𝑢)) | |
18 | reseq1 4720 | . . . . . . 7 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑢)) | |
19 | 18 | fveq2d 5322 | . . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝐺‘(𝑓 ↾ 𝑢)) = (𝐺‘(∪ 𝐵 ↾ 𝑢))) |
20 | 17, 19 | eqeq12d 2103 | . . . . 5 ⊢ (𝑓 = ∪ 𝐵 → ((𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))) |
21 | 20 | ralbidv 2381 | . . . 4 ⊢ (𝑓 = ∪ 𝐵 → (∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))) |
22 | 16, 21 | anbi12d 458 | . . 3 ⊢ (𝑓 = ∪ 𝐵 → ((𝑓 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢))) ↔ (∪ 𝐵 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))))) |
23 | 22 | spcegv 2708 | . 2 ⊢ (∪ 𝐵 ∈ V → ((∪ 𝐵 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))) → ∃𝑓(𝑓 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢))))) |
24 | 12, 15, 23 | sylc 62 | 1 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 925 = wceq 1290 ∃wex 1427 ∈ wcel 1439 {cab 2075 ∀wral 2360 ∃wrex 2361 Vcvv 2620 ∪ cun 2998 {csn 3450 〈cop 3453 ∪ cuni 3659 Ord word 4198 suc csuc 4201 ↾ cres 4454 Fun wfun 5022 Fn wfn 5023 ‘cfv 5028 recscrecs 6083 |
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 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-iord 4202 df-on 4204 df-suc 4207 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-recs 6084 |
This theorem is referenced by: tfr1onlemaccex 6127 |
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