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Theorem tfrlemisucfn 6348
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6356. (Contributed by Jim Kingdon, 2-Jul-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlemisucfn.2 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
tfrlemisucfn.3 (𝜑𝑧 ∈ On)
tfrlemisucfn.4 (𝜑𝑔 Fn 𝑧)
tfrlemisucfn.5 (𝜑𝑔𝐴)
Assertion
Ref Expression
tfrlemisucfn (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝑧,𝐴   𝑓,𝐹,𝑔,𝑥,𝑦,𝑧   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑧,𝑓,𝑔)

Proof of Theorem tfrlemisucfn
StepHypRef Expression
1 vex 2755 . . 3 𝑧 ∈ V
21a1i 9 . 2 (𝜑𝑧 ∈ V)
3 tfrlemisucfn.2 . . . 4 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
43tfrlem3-2d 6336 . . 3 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
54simprd 114 . 2 (𝜑 → (𝐹𝑔) ∈ V)
6 tfrlemisucfn.4 . 2 (𝜑𝑔 Fn 𝑧)
7 eqid 2189 . 2 (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})
8 df-suc 4389 . 2 suc 𝑧 = (𝑧 ∪ {𝑧})
9 elirrv 4565 . . 3 ¬ 𝑧𝑧
109a1i 9 . 2 (𝜑 → ¬ 𝑧𝑧)
112, 5, 6, 7, 8, 10fnunsn 5342 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wal 1362   = wceq 1364  wcel 2160  {cab 2175  wral 2468  wrex 2469  Vcvv 2752  cun 3142  {csn 3607  cop 3610  Oncon0 4381  suc csuc 4383  cres 4646  Fun wfun 5229   Fn wfn 5230  cfv 5235
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 4192  ax-pr 4227  ax-setind 4554
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-id 4311  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243
This theorem is referenced by:  tfrlemisucaccv  6349  tfrlemibfn  6352
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