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Theorem tfr1onlemsucfn 6359
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.)
Hypotheses
Ref Expression
tfr1on.f 𝐹 = recs(𝐺)
tfr1on.g (𝜑 → Fun 𝐺)
tfr1on.x (𝜑 → Ord 𝑋)
tfr1on.ex ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
tfr1onlemsucfn.1 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
tfr1onlemsucfn.3 (𝜑𝑧𝑋)
tfr1onlemsucfn.4 (𝜑𝑔 Fn 𝑧)
tfr1onlemsucfn.5 (𝜑𝑔𝐴)
Assertion
Ref Expression
tfr1onlemsucfn (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
Distinct variable groups:   𝑓,𝐺,𝑥   𝑓,𝑋,𝑥   𝑓,𝑔   𝜑,𝑓,𝑥   𝑧,𝑓,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑔)   𝐴(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐹(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐺(𝑦,𝑧,𝑔)   𝑋(𝑦,𝑧,𝑔)

Proof of Theorem tfr1onlemsucfn
StepHypRef Expression
1 tfr1onlemsucfn.3 . . 3 (𝜑𝑧𝑋)
21elexd 2765 . 2 (𝜑𝑧 ∈ V)
3 fneq2 5320 . . . . . 6 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
43imbi1d 231 . . . . 5 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
54albidv 1835 . . . 4 (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
6 tfr1on.ex . . . . . . 7 ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
763expia 1207 . . . . . 6 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
87alrimiv 1885 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
98ralrimiva 2563 . . . 4 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
105, 9, 1rspcdva 2861 . . 3 (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
11 tfr1onlemsucfn.4 . . 3 (𝜑𝑔 Fn 𝑧)
12 fneq1 5319 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
13 fveq2 5530 . . . . . 6 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
1413eleq1d 2258 . . . . 5 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
1512, 14imbi12d 234 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
1615spv 1871 . . 3 (∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
1710, 11, 16sylc 62 . 2 (𝜑 → (𝐺𝑔) ∈ V)
18 eqid 2189 . 2 (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})
19 df-suc 4386 . 2 suc 𝑧 = (𝑧 ∪ {𝑧})
20 tfr1on.x . . . 4 (𝜑 → Ord 𝑋)
21 ordelon 4398 . . . 4 ((Ord 𝑋𝑧𝑋) → 𝑧 ∈ On)
2220, 1, 21syl2anc 411 . . 3 (𝜑𝑧 ∈ On)
23 eloni 4390 . . 3 (𝑧 ∈ On → Ord 𝑧)
24 ordirr 4556 . . 3 (Ord 𝑧 → ¬ 𝑧𝑧)
2522, 23, 243syl 17 . 2 (𝜑 → ¬ 𝑧𝑧)
262, 17, 11, 18, 19, 25fnunsn 5338 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) 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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