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Mirrors > Home > ILE Home > Th. List > tfr1onlemex | GIF version |
Description: Lemma for tfr1on 6379. (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 6374 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | uniexg 4460 | . . 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 6373 | . . 3 ⊢ (𝜑 → ∪ 𝐵 Fn 𝐷) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | tfr1onlemubacc 6375 | . . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))) |
15 | 13, 14 | jca 306 | . 2 ⊢ (𝜑 → (∪ 𝐵 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))) |
16 | fneq1 5326 | . . . 4 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 Fn 𝐷 ↔ ∪ 𝐵 Fn 𝐷)) | |
17 | fveq1 5536 | . . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝑓‘𝑢) = (∪ 𝐵‘𝑢)) | |
18 | reseq1 4922 | . . . . . . 7 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑢)) | |
19 | 18 | fveq2d 5541 | . . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝐺‘(𝑓 ↾ 𝑢)) = (𝐺‘(∪ 𝐵 ↾ 𝑢))) |
20 | 17, 19 | eqeq12d 2204 | . . . . 5 ⊢ (𝑓 = ∪ 𝐵 → ((𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))) |
21 | 20 | ralbidv 2490 | . . . 4 ⊢ (𝑓 = ∪ 𝐵 → (∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))) |
22 | 16, 21 | anbi12d 473 | . . 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 2840 | . 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 104 ∧ w3a 980 = wceq 1364 ∃wex 1503 ∈ wcel 2160 {cab 2175 ∀wral 2468 ∃wrex 2469 Vcvv 2752 ∪ cun 3142 {csn 3610 〈cop 3613 ∪ cuni 3827 Ord word 4383 suc csuc 4386 ↾ cres 4649 Fun wfun 5232 Fn wfn 5233 ‘cfv 5238 recscrecs 6333 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4136 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-id 4314 df-iord 4387 df-on 4389 df-suc 4392 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-recs 6334 |
This theorem is referenced by: tfr1onlemaccex 6377 |
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