Theorem tfr1onlembacc 6321

Description: Lemma for tfr1on 6329. Each element of 𝐵 is an acceptable function. (Contributed by Jim Kingdon, 14-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)))}
tfr1onlembacc.3	⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
tfr1onlembacc.u	⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
tfr1onlembacc.4	⊢ (𝜑 → 𝐷 ∈ 𝑋)
tfr1onlembacc.5	⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))

Assertion

Ref	Expression
tfr1onlembacc	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Distinct variable groups:   𝐴,𝑓,𝑔,ℎ,𝑥,𝑧   𝐷,𝑓,𝑔,𝑥   𝑓,𝐺,𝑥,𝑦   𝑓,𝑋,𝑥   𝜑,𝑓,𝑔,ℎ,𝑥,𝑧   𝑦,𝑔,𝑧

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

Proof of Theorem tfr1onlembacc

Step	Hyp	Ref	Expression
1		tfr1onlembacc.3	. 2 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
2		simpr3 1000	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))
3		tfr1on.f	. . . . . . . 8 ⊢ 𝐹 = recs(𝐺)
4		tfr1on.g	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺)
5	4	ad2antrr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → Fun 𝐺)
6		tfr1on.x	. . . . . . . . 9 ⊢ (𝜑 → Ord 𝑋)
7	6	ad2antrr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → Ord 𝑋)
8		tfr1on.ex	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
9	8	3adant1r 1226	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
10	9	3adant1r 1226	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
11		tfr1onlemsucfn.1	. . . . . . . 8 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
12		tfr1onlembacc.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
13	12	ad2antrr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝐷 ∈ 𝑋)
14		simplr 525	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑧 ∈ 𝐷)
15		tfr1onlembacc.u	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
16	15	adantlr 474	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
17	16	adantlr 474	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
18		simpr1 998	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑔 Fn 𝑧)
19		simpr2 999	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → 𝑔 ∈ 𝐴)
20	3, 5, 7, 10, 11, 13, 14, 17, 18, 19	tfr1onlemsucaccv 6320	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}) ∈ 𝐴)
21	2, 20	eqeltrd 2247	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐷) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))) → ℎ ∈ 𝐴)
22	21	ex 114	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → ((𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝐴))
23	22	exlimdv 1812	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝐴))
24	23	rexlimdva 2587	. . 3 ⊢ (𝜑 → (∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩})) → ℎ ∈ 𝐴))
25	24	abssdv 3221	. 2 ⊢ (𝜑 → {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))} ⊆ 𝐴)
26	1, 25	eqsstrid 3193	1 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156  ∀wral 2448  ∃wrex 2449  Vcvv 2730   ∪ cun 3119   ⊆ wss 3121  {csn 3583  ⟨cop 3586  ∪ cuni 3796  Ord word 4347  suc csuc 4350   ↾ cres 4613  Fun wfun 5192   Fn wfn 5193  ‘cfv 5198  recscrecs 6283

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206

This theorem is referenced by:  tfr1onlembfn  6323  tfr1onlemubacc  6325