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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 6309. (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 2734 | . 2 ⊢ (𝜑 → 𝑧 ∈ V) |
3 | fneq2 5271 | . . . . . 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 1811 | . . . 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 1194 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
8 | 7 | alrimiv 1861 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
9 | 8 | ralrimiva 2537 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
10 | 5, 9, 1 | rspcdva 2830 | . . 3 ⊢ (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)) |
11 | tfr1onlemsucfn.4 | . . 3 ⊢ (𝜑 → 𝑔 Fn 𝑧) | |
12 | fneq1 5270 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑧 ↔ 𝑔 Fn 𝑧)) | |
13 | fveq2 5480 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔)) | |
14 | 13 | eleq1d 2233 | . . . . 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 1847 | . . 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 2164 | . 2 ⊢ (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) = (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) | |
19 | df-suc 4343 | . 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧}) | |
20 | tfr1on.x | . . . 4 ⊢ (𝜑 → Ord 𝑋) | |
21 | ordelon 4355 | . . . 4 ⊢ ((Ord 𝑋 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On) | |
22 | 20, 1, 21 | syl2anc 409 | . . 3 ⊢ (𝜑 → 𝑧 ∈ On) |
23 | eloni 4347 | . . 3 ⊢ (𝑧 ∈ On → Ord 𝑧) | |
24 | ordirr 4513 | . . 3 ⊢ (Ord 𝑧 → ¬ 𝑧 ∈ 𝑧) | |
25 | 22, 23, 24 | 3syl 17 | . 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧) |
26 | 2, 17, 11, 18, 19, 25 | fnunsn 5289 | 1 ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) Fn suc 𝑧) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∧ w3a 967 ∀wal 1340 = wceq 1342 ∈ wcel 2135 {cab 2150 ∀wral 2442 ∃wrex 2443 Vcvv 2721 ∪ cun 3109 {csn 3570 〈cop 3573 Ord word 4334 Oncon0 4335 suc csuc 4337 ↾ cres 4600 Fun wfun 5176 Fn wfn 5177 ‘cfv 5182 recscrecs 6263 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 |
This theorem is referenced by: tfr1onlemsucaccv 6300 tfr1onlembfn 6303 |
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