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Mirrors > Home > ILE Home > Th. List > tfr1onlemsucfn | GIF version |
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6326. (Contributed by Jim Kingdon, 12-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)))} |
tfr1onlemsucfn.3 | ⊢ (𝜑 → 𝑧 ∈ 𝑋) |
tfr1onlemsucfn.4 | ⊢ (𝜑 → 𝑔 Fn 𝑧) |
tfr1onlemsucfn.5 | ⊢ (𝜑 → 𝑔 ∈ 𝐴) |
Ref | Expression |
---|---|
tfr1onlemsucfn | ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) Fn suc 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfr1onlemsucfn.3 | . . 3 ⊢ (𝜑 → 𝑧 ∈ 𝑋) | |
2 | 1 | elexd 2743 | . 2 ⊢ (𝜑 → 𝑧 ∈ V) |
3 | fneq2 5285 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑓 Fn 𝑥 ↔ 𝑓 Fn 𝑧)) | |
4 | 3 | imbi1d 230 | . . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V))) |
5 | 4 | albidv 1817 | . . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V))) |
6 | tfr1on.ex | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V) | |
7 | 6 | 3expia 1200 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
8 | 7 | alrimiv 1867 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
9 | 8 | ralrimiva 2543 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
10 | 5, 9, 1 | rspcdva 2839 | . . 3 ⊢ (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)) |
11 | tfr1onlemsucfn.4 | . . 3 ⊢ (𝜑 → 𝑔 Fn 𝑧) | |
12 | fneq1 5284 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑧 ↔ 𝑔 Fn 𝑧)) | |
13 | fveq2 5494 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔)) | |
14 | 13 | eleq1d 2239 | . . . . 5 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ V ↔ (𝐺‘𝑔) ∈ V)) |
15 | 12, 14 | imbi12d 233 | . . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V))) |
16 | 15 | spv 1853 | . . 3 ⊢ (∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V)) |
17 | 10, 11, 16 | sylc 62 | . 2 ⊢ (𝜑 → (𝐺‘𝑔) ∈ V) |
18 | eqid 2170 | . 2 ⊢ (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) = (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) | |
19 | df-suc 4354 | . 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧}) | |
20 | tfr1on.x | . . . 4 ⊢ (𝜑 → Ord 𝑋) | |
21 | ordelon 4366 | . . . 4 ⊢ ((Ord 𝑋 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On) | |
22 | 20, 1, 21 | syl2anc 409 | . . 3 ⊢ (𝜑 → 𝑧 ∈ On) |
23 | eloni 4358 | . . 3 ⊢ (𝑧 ∈ On → Ord 𝑧) | |
24 | ordirr 4524 | . . 3 ⊢ (Ord 𝑧 → ¬ 𝑧 ∈ 𝑧) | |
25 | 22, 23, 24 | 3syl 17 | . 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧) |
26 | 2, 17, 11, 18, 19, 25 | fnunsn 5303 | 1 ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) Fn suc 𝑧) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∧ w3a 973 ∀wal 1346 = wceq 1348 ∈ wcel 2141 {cab 2156 ∀wral 2448 ∃wrex 2449 Vcvv 2730 ∪ cun 3119 {csn 3581 〈cop 3584 Ord word 4345 Oncon0 4346 suc csuc 4348 ↾ cres 4611 Fun wfun 5190 Fn wfn 5191 ‘cfv 5196 recscrecs 6280 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-tr 4086 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fn 5199 df-fv 5204 |
This theorem is referenced by: tfr1onlemsucaccv 6317 tfr1onlembfn 6320 |
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