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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 6247. (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 2699 | . 2 ⊢ (𝜑 → 𝑧 ∈ V) |
3 | fneq2 5212 | . . . . . 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 1796 | . . . 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 1183 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
8 | 7 | alrimiv 1846 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
9 | 8 | ralrimiva 2505 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
10 | 5, 9, 1 | rspcdva 2794 | . . 3 ⊢ (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)) |
11 | tfr1onlemsucfn.4 | . . 3 ⊢ (𝜑 → 𝑔 Fn 𝑧) | |
12 | fneq1 5211 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑧 ↔ 𝑔 Fn 𝑧)) | |
13 | fveq2 5421 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔)) | |
14 | 13 | eleq1d 2208 | . . . . 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 1832 | . . 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 2139 | . 2 ⊢ (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) = (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) | |
19 | df-suc 4293 | . 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧}) | |
20 | tfr1on.x | . . . 4 ⊢ (𝜑 → Ord 𝑋) | |
21 | ordelon 4305 | . . . 4 ⊢ ((Ord 𝑋 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On) | |
22 | 20, 1, 21 | syl2anc 408 | . . 3 ⊢ (𝜑 → 𝑧 ∈ On) |
23 | eloni 4297 | . . 3 ⊢ (𝑧 ∈ On → Ord 𝑧) | |
24 | ordirr 4457 | . . 3 ⊢ (Ord 𝑧 → ¬ 𝑧 ∈ 𝑧) | |
25 | 22, 23, 24 | 3syl 17 | . 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧) |
26 | 2, 17, 11, 18, 19, 25 | fnunsn 5230 | 1 ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) Fn suc 𝑧) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∧ w3a 962 ∀wal 1329 = wceq 1331 ∈ wcel 1480 {cab 2125 ∀wral 2416 ∃wrex 2417 Vcvv 2686 ∪ cun 3069 {csn 3527 〈cop 3530 Ord word 4284 Oncon0 4285 suc csuc 4287 ↾ cres 4541 Fun wfun 5117 Fn wfn 5118 ‘cfv 5123 recscrecs 6201 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: tfr1onlemsucaccv 6238 tfr1onlembfn 6241 |
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