Theorem tfrlemisucfn 5961

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 5969. (Contributed by Jim Kingdon, 2-Jul-2019.)

Hypotheses

Ref	Expression
tfrlemisucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
tfrlemisucfn.2	⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
tfrlemisucfn.3	⊢ (𝜑 → 𝑧 ∈ On)
tfrlemisucfn.4	⊢ (𝜑 → 𝑔 Fn 𝑧)
tfrlemisucfn.5	⊢ (𝜑 → 𝑔 ∈ 𝐴)

Assertion

Ref	Expression
tfrlemisucfn	⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) 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

Step	Hyp	Ref	Expression
1		vex 2604	. . 3 ⊢ 𝑧 ∈ V
2	1	a1i 9	. 2 ⊢ (𝜑 → 𝑧 ∈ V)
3		tfrlemisucfn.2	. . . 4 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
4	3	tfrlem3-2d 5951	. . 3 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
5	4	simprd 112	. 2 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
6		tfrlemisucfn.4	. 2 ⊢ (𝜑 → 𝑔 Fn 𝑧)
7		eqid 2081	. 2 ⊢ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})
8		df-suc 4126	. 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
9		elirrv 4291	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
10	9	a1i 9	. 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧)
11	2, 5, 6, 7, 8, 10	fnunsn 5026	1 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102  ∀wal 1282   = wceq 1284   ∈ wcel 1433  {cab 2067  ∀wral 2348  ∃wrex 2349  Vcvv 2601   ∪ cun 2971  {csn 3398  ⟨cop 3401  Oncon0 4118  suc csuc 4120   ↾ cres 4365  Fun wfun 4916   Fn wfn 4917  ‘cfv 4922

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by:  tfrlemisucaccv  5962  tfrlemibfn  5965