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Theorem tfr1onlemsucfn 6245
 Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6255. (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 2702 . 2 (𝜑𝑧 ∈ V)
3 fneq2 5220 . . . . . 6 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
43imbi1d 230 . . . . 5 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
54albidv 1797 . . . 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 1184 . . . . . 6 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
87alrimiv 1847 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
98ralrimiva 2508 . . . 4 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
105, 9, 1rspcdva 2798 . . 3 (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
11 tfr1onlemsucfn.4 . . 3 (𝜑𝑔 Fn 𝑧)
12 fneq1 5219 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
13 fveq2 5429 . . . . . 6 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
1413eleq1d 2209 . . . . 5 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
1512, 14imbi12d 233 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
1615spv 1833 . . 3 (∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
1710, 11, 16sylc 62 . 2 (𝜑 → (𝐺𝑔) ∈ V)
18 eqid 2140 . 2 (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})
19 df-suc 4301 . 2 suc 𝑧 = (𝑧 ∪ {𝑧})
20 tfr1on.x . . . 4 (𝜑 → Ord 𝑋)
21 ordelon 4313 . . . 4 ((Ord 𝑋𝑧𝑋) → 𝑧 ∈ On)
2220, 1, 21syl2anc 409 . . 3 (𝜑𝑧 ∈ On)
23 eloni 4305 . . 3 (𝑧 ∈ On → Ord 𝑧)
24 ordirr 4465 . . 3 (Ord 𝑧 → ¬ 𝑧𝑧)
2522, 23, 243syl 17 . 2 (𝜑 → ¬ 𝑧𝑧)
262, 17, 11, 18, 19, 25fnunsn 5238 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
 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   Fn wfn 5126  ‘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-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139 This theorem is referenced by:  tfr1onlemsucaccv  6246  tfr1onlembfn  6249
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