Theorem tfr1onlemsucfn 6343

Description: We can extend an acceptable function by one element to produce a function. Lemma for tfr1on 6353. (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 2752	. 2 ⊢ (𝜑 → 𝑧 ∈ V)
3		fneq2 5307	. . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑓 Fn 𝑥 ↔ 𝑓 Fn 𝑧))
4	3	imbi1d 231	. . . . 5 ⊢ (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V)))
5	4	albidv 1824	. . . 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 1205	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
8	7	alrimiv 1874	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
9	8	ralrimiva 2550	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑓(𝑓 Fn 𝑥 → (𝐺‘𝑓) ∈ V))
10	5, 9, 1	rspcdva 2848	. . 3 ⊢ (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V))
11		tfr1onlemsucfn.4	. . 3 ⊢ (𝜑 → 𝑔 Fn 𝑧)
12		fneq1 5306	. . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓 Fn 𝑧 ↔ 𝑔 Fn 𝑧))
13		fveq2 5517	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝐺‘𝑓) = (𝐺‘𝑔))
14	13	eleq1d 2246	. . . . 5 ⊢ (𝑓 = 𝑔 → ((𝐺‘𝑓) ∈ V ↔ (𝐺‘𝑔) ∈ V))
15	12, 14	imbi12d 234	. . . 4 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺‘𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺‘𝑔) ∈ V)))
16	15	spv 1860	. . 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 2177	. 2 ⊢ (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})
19		df-suc 4373	. 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧})
20		tfr1on.x	. . . 4 ⊢ (𝜑 → Ord 𝑋)
21		ordelon 4385	. . . 4 ⊢ ((Ord 𝑋 ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On)
22	20, 1, 21	syl2anc 411	. . 3 ⊢ (𝜑 → 𝑧 ∈ On)
23		eloni 4377	. . 3 ⊢ (𝑧 ∈ On → Ord 𝑧)
24		ordirr 4543	. . 3 ⊢ (Ord 𝑧 → ¬ 𝑧 ∈ 𝑧)
25	22, 23, 24	3syl 17	. 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧)
26	2, 17, 11, 18, 19, 25	fnunsn 5325	1 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) Fn suc 𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∧ w3a 978  ∀wal 1351   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  Vcvv 2739   ∪ cun 3129  {csn 3594  ⟨cop 3597  Ord word 4364  Oncon0 4365  suc csuc 4367   ↾ cres 4630  Fun wfun 5212   Fn wfn 5213  ‘cfv 5218  recscrecs 6307

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by:  tfr1onlemsucaccv  6344  tfr1onlembfn  6347