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Theorem tfr1onlemsucfn 6316
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6326. (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 2743 . 2 (𝜑𝑧 ∈ V)
3 fneq2 5285 . . . . . 6 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
43imbi1d 230 . . . . 5 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
54albidv 1817 . . . 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 1200 . . . . . 6 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
87alrimiv 1867 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
98ralrimiva 2543 . . . 4 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
105, 9, 1rspcdva 2839 . . 3 (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
11 tfr1onlemsucfn.4 . . 3 (𝜑𝑔 Fn 𝑧)
12 fneq1 5284 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
13 fveq2 5494 . . . . . 6 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
1413eleq1d 2239 . . . . 5 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
1512, 14imbi12d 233 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
1615spv 1853 . . 3 (∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
1710, 11, 16sylc 62 . 2 (𝜑 → (𝐺𝑔) ∈ V)
18 eqid 2170 . 2 (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})
19 df-suc 4354 . 2 suc 𝑧 = (𝑧 ∪ {𝑧})
20 tfr1on.x . . . 4 (𝜑 → Ord 𝑋)
21 ordelon 4366 . . . 4 ((Ord 𝑋𝑧𝑋) → 𝑧 ∈ On)
2220, 1, 21syl2anc 409 . . 3 (𝜑𝑧 ∈ On)
23 eloni 4358 . . 3 (𝑧 ∈ On → Ord 𝑧)
24 ordirr 4524 . . 3 (Ord 𝑧 → ¬ 𝑧𝑧)
2522, 23, 243syl 17 . 2 (𝜑 → ¬ 𝑧𝑧)
262, 17, 11, 18, 19, 25fnunsn 5303 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  w3a 973  wal 1346   = wceq 1348  wcel 2141  {cab 2156  wral 2448  wrex 2449  Vcvv 2730  cun 3119  {csn 3581  cop 3584  Ord word 4345  Oncon0 4346  suc csuc 4348  cres 4611  Fun wfun 5190   Fn wfn 5191  cfv 5196  recscrecs 6280
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-tr 4086  df-id 4276  df-iord 4349  df-on 4351  df-suc 4354  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fn 5199  df-fv 5204
This theorem is referenced by:  tfr1onlemsucaccv  6317  tfr1onlembfn  6320
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