Theorem tfr1onlemex 6432

Description: Lemma for tfr1on 6435. (Contributed by Jim Kingdon, 16-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
tfr1onlemex	⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢))))

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   𝐵,𝑓,𝑔,ℎ,𝑤,𝑧   𝑢,𝐵,𝑓,𝑤   𝐷,ℎ,𝑤,𝑧,𝑥   𝑢,𝐷   ℎ,𝐺,𝑧,𝑦   𝑢,𝐺,𝑤   𝑔,𝑋,𝑧   𝜑,𝑤   𝑦,𝑤

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

Proof of Theorem tfr1onlemex

Step	Hyp	Ref	Expression
1		tfr1on.f	. . . 4 ⊢ 𝐹 = recs(𝐺)
2		tfr1on.g	. . . 4 ⊢ (𝜑 → Fun 𝐺)
3		tfr1on.x	. . . 4 ⊢ (𝜑 → Ord 𝑋)
4		tfr1on.ex	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓 Fn 𝑥) → (𝐺‘𝑓) ∈ V)
5		tfr1onlemsucfn.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
6		tfr1onlembacc.3	. . . 4 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
7		tfr1onlembacc.u	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
8		tfr1onlembacc.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
9		tfr1onlembacc.5	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlembex 6430	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
11		uniexg 4485	. . 3 ⊢ (𝐵 ∈ V → ∪ 𝐵 ∈ V)
12	10, 11	syl 14	. 2 ⊢ (𝜑 → ∪ 𝐵 ∈ V)
13	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlembfn 6429	. . 3 ⊢ (𝜑 → ∪ 𝐵 Fn 𝐷)
14	1, 2, 3, 4, 5, 6, 7, 8, 9	tfr1onlemubacc 6431	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))
15	13, 14	jca 306	. 2 ⊢ (𝜑 → (∪ 𝐵 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))))
16		fneq1 5361	. . . 4 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 Fn 𝐷 ↔ ∪ 𝐵 Fn 𝐷))
17		fveq1 5574	. . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝑓‘𝑢) = (∪ 𝐵‘𝑢))
18		reseq1 4952	. . . . . . 7 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑢))
19	18	fveq2d 5579	. . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝐺‘(𝑓 ↾ 𝑢)) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))
20	17, 19	eqeq12d 2219	. . . . 5 ⊢ (𝑓 = ∪ 𝐵 → ((𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))))
21	20	ralbidv 2505	. . . 4 ⊢ (𝑓 = ∪ 𝐵 → (∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))))
22	16, 21	anbi12d 473	. . 3 ⊢ (𝑓 = ∪ 𝐵 → ((𝑓 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢))) ↔ (∪ 𝐵 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))))
23	22	spcegv 2860	. 2 ⊢ (∪ 𝐵 ∈ V → ((∪ 𝐵 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))) → ∃𝑓(𝑓 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)))))
24	12, 15, 23	sylc 62	1 ⊢ (𝜑 → ∃𝑓(𝑓 Fn 𝐷 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢))))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1372  ∃wex 1514   ∈ wcel 2175  {cab 2190  ∀wral 2483  ∃wrex 2484  Vcvv 2771   ∪ cun 3163  {csn 3632  ⟨cop 3635  ∪ cuni 3849  Ord word 4408  suc csuc 4411   ↾ cres 4676  Fun wfun 5264   Fn wfn 5265  ‘cfv 5270  recscrecs 6389

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-recs 6390

This theorem is referenced by:  tfr1onlemaccex  6433