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Theorem tfrlemisucfn 5994
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6002. (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 2613 . . 3 𝑧 ∈ V
21a1i 9 . 2 (𝜑𝑧 ∈ V)
3 tfrlemisucfn.2 . . . 4 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
43tfrlem3-2d 5982 . . 3 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
54simprd 112 . 2 (𝜑 → (𝐹𝑔) ∈ V)
6 tfrlemisucfn.4 . 2 (𝜑𝑔 Fn 𝑧)
7 eqid 2083 . 2 (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})
8 df-suc 4155 . 2 suc 𝑧 = (𝑧 ∪ {𝑧})
9 elirrv 4320 . . 3 ¬ 𝑧𝑧
109a1i 9 . 2 (𝜑 → ¬ 𝑧𝑧)
112, 5, 6, 7, 8, 10fnunsn 5058 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wal 1283   = wceq 1285  wcel 1434  {cab 2069  wral 2353  wrex 2354  Vcvv 2610  cun 2981  {csn 3417  cop 3420  Oncon0 4147  suc csuc 4149  cres 4394  Fun wfun 4947   Fn wfn 4948  cfv 4953
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-setind 4309
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-id 4077  df-suc 4155  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-iota 4918  df-fun 4955  df-fn 4956  df-fv 4961
This theorem is referenced by:  tfrlemisucaccv  5995  tfrlemibfn  5998
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