Theorem tfrlemisucfn 6410

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6418. (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 2775	. . 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 6398	. . 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 2205	. 2 ⊢ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})
8		df-suc 4418	. 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
9		elirrv 4596	. . 3 ⊢ ¬ 𝑧 ∈ 𝑧
10	9	a1i 9	. 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧)
11	2, 5, 6, 7, 8, 10	fnunsn 5383	1 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1371   = wceq 1373   ∈ wcel 2176  {cab 2191  ∀wral 2484  ∃wrex 2485  Vcvv 2772   ∪ cun 3164  {csn 3633  ⟨cop 3636  Oncon0 4410  suc csuc 4412   ↾ cres 4677  Fun wfun 5265   Fn wfn 5266  ‘cfv 5271

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279

This theorem is referenced by:  tfrlemisucaccv  6411  tfrlemibfn  6414