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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 6369. (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 2765 | . 2 ⊢ (𝜑 → 𝑧 ∈ V) |
3 | fneq2 5320 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑓 Fn 𝑥 ↔ 𝑓 Fn 𝑧)) | |
4 | 3 | imbi1d 231 | . . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V))) |
5 | 4 | albidv 1835 | . . . 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 1207 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
8 | 7 | alrimiv 1885 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
9 | 8 | ralrimiva 2563 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
10 | 5, 9, 1 | rspcdva 2861 | . . 3 ⊢ (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)) |
11 | tfr1onlemsucfn.4 | . . 3 ⊢ (𝜑 → 𝑔 Fn 𝑧) | |
12 | fneq1 5319 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑧 ↔ 𝑔 Fn 𝑧)) | |
13 | fveq2 5530 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔)) | |
14 | 13 | eleq1d 2258 | . . . . 5 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ V ↔ (𝐺‘𝑔) ∈ V)) |
15 | 12, 14 | imbi12d 234 | . . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V))) |
16 | 15 | spv 1871 | . . 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 2189 | . 2 ⊢ (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) = (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) | |
19 | df-suc 4386 | . 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧}) | |
20 | tfr1on.x | . . . 4 ⊢ (𝜑 → Ord 𝑋) | |
21 | ordelon 4398 | . . . 4 ⊢ ((Ord 𝑋 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On) | |
22 | 20, 1, 21 | syl2anc 411 | . . 3 ⊢ (𝜑 → 𝑧 ∈ On) |
23 | eloni 4390 | . . 3 ⊢ (𝑧 ∈ On → Ord 𝑧) | |
24 | ordirr 4556 | . . 3 ⊢ (Ord 𝑧 → ¬ 𝑧 ∈ 𝑧) | |
25 | 22, 23, 24 | 3syl 17 | . 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧) |
26 | 2, 17, 11, 18, 19, 25 | fnunsn 5338 | 1 ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) Fn suc 𝑧) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∧ w3a 980 ∀wal 1362 = wceq 1364 ∈ wcel 2160 {cab 2175 ∀wral 2468 ∃wrex 2469 Vcvv 2752 ∪ cun 3142 {csn 3607 〈cop 3610 Ord word 4377 Oncon0 4378 suc csuc 4380 ↾ cres 4643 Fun wfun 5225 Fn wfn 5226 ‘cfv 5231 recscrecs 6323 |
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-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-setind 4551 |
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-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-suc 4386 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-iota 5193 df-fun 5233 df-fn 5234 df-fv 5239 |
This theorem is referenced by: tfr1onlemsucaccv 6360 tfr1onlembfn 6363 |
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