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Mirrors > Home > ILE Home > Th. List > tfrcllemsucfn | GIF version |
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrcl 6269. (Contributed by Jim Kingdon, 24-Mar-2022.) |
Ref | Expression |
---|---|
tfrcl.f | ⊢ 𝐹 = recs(𝐺) |
tfrcl.g | ⊢ (𝜑 → Fun 𝐺) |
tfrcl.x | ⊢ (𝜑 → Ord 𝑋) |
tfrcl.ex | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆) |
tfrcllemsucfn.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} |
tfrcllemsucfn.3 | ⊢ (𝜑 → 𝑧 ∈ 𝑋) |
tfrcllemsucfn.4 | ⊢ (𝜑 → 𝑔:𝑧⟶𝑆) |
tfrcllemsucfn.5 | ⊢ (𝜑 → 𝑔 ∈ 𝐴) |
Ref | Expression |
---|---|
tfrcllemsucfn | ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}):suc 𝑧⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrcllemsucfn.4 | . . 3 ⊢ (𝜑 → 𝑔:𝑧⟶𝑆) | |
2 | tfrcllemsucfn.3 | . . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝑋) | |
3 | 2 | elexd 2702 | . . 3 ⊢ (𝜑 → 𝑧 ∈ V) |
4 | tfrcl.x | . . . . 5 ⊢ (𝜑 → Ord 𝑋) | |
5 | ordelon 4313 | . . . . 5 ⊢ ((Ord 𝑋 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On) | |
6 | 4, 2, 5 | syl2anc 409 | . . . 4 ⊢ (𝜑 → 𝑧 ∈ On) |
7 | eloni 4305 | . . . 4 ⊢ (𝑧 ∈ On → Ord 𝑧) | |
8 | ordirr 4465 | . . . 4 ⊢ (Ord 𝑧 → ¬ 𝑧 ∈ 𝑧) | |
9 | 6, 7, 8 | 3syl 17 | . . 3 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧) |
10 | feq2 5264 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑓:𝑥⟶𝑆 ↔ 𝑓:𝑧⟶𝑆)) | |
11 | 10 | imbi1d 230 | . . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))) |
12 | 11 | albidv 1797 | . . . . 5 ⊢ (𝑥 = 𝑧 → (∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ ∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆))) |
13 | tfrcl.ex | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆) | |
14 | 13 | 3expia 1184 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)) |
15 | 14 | alrimiv 1847 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)) |
16 | 15 | ralrimiva 2508 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)) |
17 | 12, 16, 2 | rspcdva 2798 | . . . 4 ⊢ (𝜑 → ∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)) |
18 | feq1 5263 | . . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑓:𝑧⟶𝑆 ↔ 𝑔:𝑧⟶𝑆)) | |
19 | fveq2 5429 | . . . . . . 7 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔)) | |
20 | 19 | eleq1d 2209 | . . . . . 6 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ 𝑆 ↔ (𝐺‘𝑔) ∈ 𝑆)) |
21 | 18, 20 | imbi12d 233 | . . . . 5 ⊢ (𝑓 = 𝑔 → ((𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆))) |
22 | 21 | spv 1833 | . . . 4 ⊢ (∀𝑓(𝑓:𝑧⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) → (𝑔:𝑧⟶𝑆 → (𝐺‘𝑔) ∈ 𝑆)) |
23 | 17, 1, 22 | sylc 62 | . . 3 ⊢ (𝜑 → (𝐺‘𝑔) ∈ 𝑆) |
24 | fsnunf 5628 | . . 3 ⊢ ((𝑔:𝑧⟶𝑆 ∧ (𝑧 ∈ V ∧ ¬ 𝑧 ∈ 𝑧) ∧ (𝐺‘𝑔) ∈ 𝑆) → (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}):(𝑧 ∪ {𝑧})⟶𝑆) | |
25 | 1, 3, 9, 23, 24 | syl121anc 1222 | . 2 ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}):(𝑧 ∪ {𝑧})⟶𝑆) |
26 | df-suc 4301 | . . 3 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧}) | |
27 | 26 | feq2i 5274 | . 2 ⊢ ((𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}):suc 𝑧⟶𝑆 ↔ (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}):(𝑧 ∪ {𝑧})⟶𝑆) |
28 | 25, 27 | sylibr 133 | 1 ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐺‘𝑔)〉}):suc 𝑧⟶𝑆) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∧ w3a 963 ∀wal 1330 = wceq 1332 ∈ wcel 1481 {cab 2126 ∀wral 2417 ∃wrex 2418 Vcvv 2689 ∪ cun 3074 {csn 3532 〈cop 3535 Ord word 4292 Oncon0 4293 suc csuc 4295 ↾ cres 4549 Fun wfun 5125 ⟶wf 5127 ‘cfv 5131 recscrecs 6209 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 |
This theorem is referenced by: tfrcllemsucaccv 6259 tfrcllembfn 6262 |
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