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Theorem tfrlemisucfn 5938
 Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 5946. (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 2560 . . 3 𝑧 ∈ V
21a1i 9 . 2 (𝜑𝑧 ∈ V)
3 tfrlemisucfn.2 . . . 4 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
43tfrlem3-2d 5928 . . 3 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
54simprd 107 . 2 (𝜑 → (𝐹𝑔) ∈ V)
6 tfrlemisucfn.4 . 2 (𝜑𝑔 Fn 𝑧)
7 eqid 2040 . 2 (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})
8 df-suc 4108 . 2 suc 𝑧 = (𝑧 ∪ {𝑧})
9 elirrv 4272 . . 3 ¬ 𝑧𝑧
109a1i 9 . 2 (𝜑 → ¬ 𝑧𝑧)
112, 5, 6, 7, 8, 10fnunsn 5006 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  Vcvv 2557   ∪ cun 2915  {csn 3375  ⟨cop 3378  Oncon0 4100  suc csuc 4102   ↾ cres 4347  Fun wfun 4896   Fn wfn 4897  ‘cfv 4902 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910 This theorem is referenced by:  tfrlemisucaccv  5939  tfrlemibfn  5942
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