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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 6201. (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 2670 | . 2 ⊢ (𝜑 → 𝑧 ∈ V) |
3 | fneq2 5170 | . . . . . 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 1778 | . . . 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 1166 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
8 | 7 | alrimiv 1828 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
9 | 8 | ralrimiva 2479 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V)) |
10 | 5, 9, 1 | rspcdva 2765 | . . 3 ⊢ (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)) |
11 | tfr1onlemsucfn.4 | . . 3 ⊢ (𝜑 → 𝑔 Fn 𝑧) | |
12 | fneq1 5169 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑧 ↔ 𝑔 Fn 𝑧)) | |
13 | fveq2 5375 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔)) | |
14 | 13 | eleq1d 2183 | . . . . 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 1814 | . . 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 2115 | . 2 ⊢ (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) = (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) | |
19 | df-suc 4253 | . 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧}) | |
20 | tfr1on.x | . . . 4 ⊢ (𝜑 → Ord 𝑋) | |
21 | ordelon 4265 | . . . 4 ⊢ ((Ord 𝑋 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On) | |
22 | 20, 1, 21 | syl2anc 406 | . . 3 ⊢ (𝜑 → 𝑧 ∈ On) |
23 | eloni 4257 | . . 3 ⊢ (𝑧 ∈ On → Ord 𝑧) | |
24 | ordirr 4417 | . . 3 ⊢ (Ord 𝑧 → ¬ 𝑧 ∈ 𝑧) | |
25 | 22, 23, 24 | 3syl 17 | . 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧) |
26 | 2, 17, 11, 18, 19, 25 | fnunsn 5188 | 1 ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}) Fn suc 𝑧) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∧ w3a 945 ∀wal 1312 = wceq 1314 ∈ wcel 1463 {cab 2101 ∀wral 2390 ∃wrex 2391 Vcvv 2657 ∪ cun 3035 {csn 3493 〈cop 3496 Ord word 4244 Oncon0 4245 suc csuc 4247 ↾ cres 4501 Fun wfun 5075 Fn wfn 5076 ‘cfv 5081 recscrecs 6155 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-setind 4412 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-v 2659 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-tr 3987 df-id 4175 df-iord 4248 df-on 4250 df-suc 4253 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-iota 5046 df-fun 5083 df-fn 5084 df-fv 5089 |
This theorem is referenced by: tfr1onlemsucaccv 6192 tfr1onlembfn 6195 |
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