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Theorem tfrlemisucfn 6071
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6079. (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 2622 . . 3 𝑧 ∈ V
21a1i 9 . 2 (𝜑𝑧 ∈ V)
3 tfrlemisucfn.2 . . . 4 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
43tfrlem3-2d 6059 . . 3 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
54simprd 112 . 2 (𝜑 → (𝐹𝑔) ∈ V)
6 tfrlemisucfn.4 . 2 (𝜑𝑔 Fn 𝑧)
7 eqid 2088 . 2 (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})
8 df-suc 4189 . 2 suc 𝑧 = (𝑧 ∪ {𝑧})
9 elirrv 4354 . . 3 ¬ 𝑧𝑧
109a1i 9 . 2 (𝜑 → ¬ 𝑧𝑧)
112, 5, 6, 7, 8, 10fnunsn 5107 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1287   = wceq 1289  wcel 1438  {cab 2074  wral 2359  wrex 2360  Vcvv 2619  cun 2995  {csn 3441  cop 3444  Oncon0 4181  suc csuc 4183  cres 4430  Fun wfun 4996   Fn wfn 4997  cfv 5002
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-id 4111  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:  tfrlemisucaccv  6072  tfrlemibfn  6075
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