Theorem tfr1onlemsucfn 6237

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6247. (Contributed by Jim Kingdon, 12-Mar-2022.)

Hypotheses

Ref	Expression
tfr1on.f	⊢ 𝐹 = recs(𝐺)
tfr1on.g	⊢ (𝜑 → Fun 𝐺)
tfr1on.x	⊢ (𝜑 → Ord 𝑋)
tfr1on.ex	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
tfr1onlemsucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
tfr1onlemsucfn.3	⊢ (𝜑 → 𝑧 ∈ 𝑋)
tfr1onlemsucfn.4	⊢ (𝜑 → 𝑔 Fn 𝑧)
tfr1onlemsucfn.5	⊢ (𝜑 → 𝑔 ∈ 𝐴)

Assertion

Ref	Expression
tfr1onlemsucfn	⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) Fn suc 𝑧)

Distinct variable groups:   𝑓,𝐺,𝑥   𝑓,𝑋,𝑥   𝑓,𝑔   𝜑,𝑓,𝑥   𝑧,𝑓,𝑥

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑔)   𝐴(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐹(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐺(𝑦,𝑧,𝑔)   𝑋(𝑦,𝑧,𝑔)

Proof of Theorem tfr1onlemsucfn

Step	Hyp	Ref	Expression
1		tfr1onlemsucfn.3	. . 3 ⊢ (𝜑 → 𝑧 ∈ 𝑋)
2	1	elexd 2699	. 2 ⊢ (𝜑 → 𝑧 ∈ V)
3		fneq2 5212	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑓 Fn 𝑥 ↔ 𝑓 Fn 𝑧))
4	3	imbi1d 230	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)))
5	4	albidv 1796	. . . 4 ⊢ (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)))
6		tfr1on.ex	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
7	6	3expia 1183	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
8	7	alrimiv 1846	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
9	8	ralrimiva 2505	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
10	5, 9, 1	rspcdva 2794	. . 3 ⊢ (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V))
11		tfr1onlemsucfn.4	. . 3 ⊢ (𝜑 → 𝑔 Fn 𝑧)
12		fneq1 5211	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑧 ↔ 𝑔 Fn 𝑧))
13		fveq2 5421	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
14	13	eleq1d 2208	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ V ↔ (𝐺‘𝑔) ∈ V))
15	12, 14	imbi12d 233	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V)))
16	15	spv 1832	. . 3 ⊢ (∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V))
17	10, 11, 16	sylc 62	. 2 ⊢ (𝜑 → (𝐺‘𝑔) ∈ V)
18		eqid 2139	. 2 ⊢ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})
19		df-suc 4293	. 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
20		tfr1on.x	. . . 4 ⊢ (𝜑 → Ord 𝑋)
21		ordelon 4305	. . . 4 ⊢ ((Ord 𝑋 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On)
22	20, 1, 21	syl2anc 408	. . 3 ⊢ (𝜑 → 𝑧 ∈ On)
23		eloni 4297	. . 3 ⊢ (𝑧 ∈ On → Ord 𝑧)
24		ordirr 4457	. . 3 ⊢ (Ord 𝑧 → ¬ 𝑧 ∈ 𝑧)
25	22, 23, 24	3syl 17	. 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧)
26	2, 17, 11, 18, 19, 25	fnunsn 5230	1 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) Fn suc 𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∧ w3a 962  ∀wal 1329   = wceq 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∃wrex 2417  Vcvv 2686   ∪ cun 3069  {csn 3527  ⟨cop 3530  Ord word 4284  Oncon0 4285  suc csuc 4287   ↾ cres 4541  Fun wfun 5117   Fn wfn 5118  ‘cfv 5123  recscrecs 6201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131

This theorem is referenced by:  tfr1onlemsucaccv  6238  tfr1onlembfn  6241