Theorem tfrlemisucfn 6533

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6541. (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 2806	. . 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 6521	. . 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 2231	. 2 ⊢ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})
8		df-suc 4474	. 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
9		elirrv 4652	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
10	9	a1i 9	. 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧)
11	2, 5, 6, 7, 8, 10	fnunsn 5446	1 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1396   = wceq 1398   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  Vcvv 2803   ∪ cun 3199  {csn 3673  ⟨cop 3676  Oncon0 4466  suc csuc 4468   ↾ cres 4733  Fun wfun 5327   Fn wfn 5328  ‘cfv 5333

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-suc 4474  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341

This theorem is referenced by:  tfrlemisucaccv  6534  tfrlemibfn  6537