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Theorem tfrlemisucfn 5968
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 5976. (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 2577 . . 3 𝑧 ∈ V
21a1i 9 . 2 (𝜑𝑧 ∈ V)
3 tfrlemisucfn.2 . . . 4 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
43tfrlem3-2d 5958 . . 3 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
54simprd 111 . 2 (𝜑 → (𝐹𝑔) ∈ V)
6 tfrlemisucfn.4 . 2 (𝜑𝑔 Fn 𝑧)
7 eqid 2056 . 2 (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})
8 df-suc 4135 . 2 suc 𝑧 = (𝑧 ∪ {𝑧})
9 elirrv 4299 . . 3 ¬ 𝑧𝑧
109a1i 9 . 2 (𝜑 → ¬ 𝑧𝑧)
112, 5, 6, 7, 8, 10fnunsn 5033 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wal 1257   = wceq 1259  wcel 1409  {cab 2042  wral 2323  wrex 2324  Vcvv 2574  cun 2942  {csn 3402  cop 3405  Oncon0 4127  suc csuc 4129  cres 4374  Fun wfun 4923   Fn wfn 4924  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-setind 4289
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-suc 4135  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937
This theorem is referenced by:  tfrlemisucaccv  5969  tfrlemibfn  5972
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