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Theorem tfr1onlemsucfn 6087
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6097. (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 2632 . 2 (𝜑𝑧 ∈ V)
3 fneq2 5089 . . . . . 6 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
43imbi1d 229 . . . . 5 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
54albidv 1752 . . . 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 1145 . . . . . 6 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
87alrimiv 1802 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
98ralrimiva 2446 . . . 4 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
105, 9, 1rspcdva 2727 . . 3 (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
11 tfr1onlemsucfn.4 . . 3 (𝜑𝑔 Fn 𝑧)
12 fneq1 5088 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
13 fveq2 5289 . . . . . 6 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
1413eleq1d 2156 . . . . 5 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
1512, 14imbi12d 232 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
1615spv 1788 . . 3 (∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
1710, 11, 16sylc 61 . 2 (𝜑 → (𝐺𝑔) ∈ V)
18 eqid 2088 . 2 (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})
19 df-suc 4189 . 2 suc 𝑧 = (𝑧 ∪ {𝑧})
20 tfr1on.x . . . 4 (𝜑 → Ord 𝑋)
21 ordelon 4201 . . . 4 ((Ord 𝑋𝑧𝑋) → 𝑧 ∈ On)
2220, 1, 21syl2anc 403 . . 3 (𝜑𝑧 ∈ On)
23 eloni 4193 . . 3 (𝑧 ∈ On → Ord 𝑧)
24 ordirr 4348 . . 3 (Ord 𝑧 → ¬ 𝑧𝑧)
2522, 23, 243syl 17 . 2 (𝜑 → ¬ 𝑧𝑧)
262, 17, 11, 18, 19, 25fnunsn 5107 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  w3a 924  wal 1287   = wceq 1289  wcel 1438  {cab 2074  wral 2359  wrex 2360  Vcvv 2619  cun 2995  {csn 3441  cop 3444  Ord word 4180  Oncon0 4181  suc csuc 4183  cres 4430  Fun wfun 4996   Fn wfn 4997  cfv 5002  recscrecs 6051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010
This theorem is referenced by:  tfr1onlemsucaccv  6088  tfr1onlembfn  6091
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