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Theorem tfr1onlemsucfn 6191
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6201. (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 2670 . 2 (𝜑𝑧 ∈ V)
3 fneq2 5170 . . . . . 6 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
43imbi1d 230 . . . . 5 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
54albidv 1778 . . . 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 1166 . . . . . 6 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
87alrimiv 1828 . . . . 5 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
98ralrimiva 2479 . . . 4 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
105, 9, 1rspcdva 2765 . . 3 (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
11 tfr1onlemsucfn.4 . . 3 (𝜑𝑔 Fn 𝑧)
12 fneq1 5169 . . . . 5 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
13 fveq2 5375 . . . . . 6 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
1413eleq1d 2183 . . . . 5 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
1512, 14imbi12d 233 . . . 4 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
1615spv 1814 . . 3 (∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
1710, 11, 16sylc 62 . 2 (𝜑 → (𝐺𝑔) ∈ V)
18 eqid 2115 . 2 (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})
19 df-suc 4253 . 2 suc 𝑧 = (𝑧 ∪ {𝑧})
20 tfr1on.x . . . 4 (𝜑 → Ord 𝑋)
21 ordelon 4265 . . . 4 ((Ord 𝑋𝑧𝑋) → 𝑧 ∈ On)
2220, 1, 21syl2anc 406 . . 3 (𝜑𝑧 ∈ On)
23 eloni 4257 . . 3 (𝑧 ∈ On → Ord 𝑧)
24 ordirr 4417 . . 3 (Ord 𝑧 → ¬ 𝑧𝑧)
2522, 23, 243syl 17 . 2 (𝜑 → ¬ 𝑧𝑧)
262, 17, 11, 18, 19, 25fnunsn 5188 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  w3a 945  wal 1312   = wceq 1314  wcel 1463  {cab 2101  wral 2390  wrex 2391  Vcvv 2657  cun 3035  {csn 3493  cop 3496  Ord word 4244  Oncon0 4245  suc csuc 4247  cres 4501  Fun wfun 5075   Fn wfn 5076  cfv 5081  recscrecs 6155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-setind 4412
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-v 2659  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fn 5084  df-fv 5089
This theorem is referenced by:  tfr1onlemsucaccv  6192  tfr1onlembfn  6195
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