Theorem tfrlemisucfn 6485

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6493. (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 2803	. . 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 6473	. . 3 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
5	4	simprd 114	. 2 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
6		tfrlemisucfn.4	. 2 ⊢ (𝜑 → 𝑔 Fn 𝑧)
7		eqid 2229	. 2 ⊢ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})
8		df-suc 4466	. 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
9		elirrv 4644	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
10	9	a1i 9	. 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧)
11	2, 5, 6, 7, 8, 10	fnunsn 5436	1 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1393   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  Vcvv 2800   ∪ cun 3196  {csn 3667  ⟨cop 3670  Oncon0 4458  suc csuc 4460   ↾ cres 4725  Fun wfun 5318   Fn wfn 5319  ‘cfv 5324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332

This theorem is referenced by:  tfrlemisucaccv  6486  tfrlemibfn  6489