Theorem tfrcllemex 6459

Description: Lemma for tfrcl 6463. (Contributed by Jim Kingdon, 26-Mar-2022.)

Hypotheses

Ref	Expression
tfrcl.f	⊢ 𝐹 = recs(𝐺)
tfrcl.g	⊢ (𝜑 → Fun 𝐺)
tfrcl.x	⊢ (𝜑 → Ord 𝑋)
tfrcl.ex	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
tfrcllemsucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
tfrcllembacc.3	⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
tfrcllembacc.u	⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
tfrcllembacc.4	⊢ (𝜑 → 𝐷 ∈ 𝑋)
tfrcllembacc.5	⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))

Assertion

Ref	Expression
tfrcllemex	⊢ (𝜑 → ∃𝑓(𝑓:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢))))

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

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

Proof of Theorem tfrcllemex

Step	Hyp	Ref	Expression
1		tfrcl.f	. . . 4 ⊢ 𝐹 = recs(𝐺)
2		tfrcl.g	. . . 4 ⊢ (𝜑 → Fun 𝐺)
3		tfrcl.x	. . . 4 ⊢ (𝜑 → Ord 𝑋)
4		tfrcl.ex	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆)
5		tfrcllemsucfn.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
6		tfrcllembacc.3	. . . 4 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐺‘𝑔)⟩}))}
7		tfrcllembacc.u	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋)
8		tfrcllembacc.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
9		tfrcllembacc.5	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐷 ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ 𝑤))))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	tfrcllembex 6457	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
11		uniexg 4494	. . 3 ⊢ (𝐵 ∈ V → ∪ 𝐵 ∈ V)
12	10, 11	syl 14	. 2 ⊢ (𝜑 → ∪ 𝐵 ∈ V)
13	1, 2, 3, 4, 5, 6, 7, 8, 9	tfrcllembfn 6456	. . 3 ⊢ (𝜑 → ∪ 𝐵:𝐷⟶𝑆)
14	1, 2, 3, 4, 5, 6, 7, 8, 9	tfrcllemubacc 6458	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))
15	13, 14	jca 306	. 2 ⊢ (𝜑 → (∪ 𝐵:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))))
16		feq1 5418	. . . 4 ⊢ (𝑓 = ∪ 𝐵 → (𝑓:𝐷⟶𝑆 ↔ ∪ 𝐵:𝐷⟶𝑆))
17		fveq1 5588	. . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝑓‘𝑢) = (∪ 𝐵‘𝑢))
18		reseq1 4962	. . . . . . 7 ⊢ (𝑓 = ∪ 𝐵 → (𝑓 ↾ 𝑢) = (∪ 𝐵 ↾ 𝑢))
19	18	fveq2d 5593	. . . . . 6 ⊢ (𝑓 = ∪ 𝐵 → (𝐺‘(𝑓 ↾ 𝑢)) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))
20	17, 19	eqeq12d 2221	. . . . 5 ⊢ (𝑓 = ∪ 𝐵 → ((𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)) ↔ (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))))
21	20	ralbidv 2507	. . . 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:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢))) ↔ (∪ 𝐵:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢)))))
23	22	spcegv 2865	. 2 ⊢ (∪ 𝐵 ∈ V → ((∪ 𝐵:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (∪ 𝐵‘𝑢) = (𝐺‘(∪ 𝐵 ↾ 𝑢))) → ∃𝑓(𝑓:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢)))))
24	12, 15, 23	sylc 62	1 ⊢ (𝜑 → ∃𝑓(𝑓:𝐷⟶𝑆 ∧ ∀𝑢 ∈ 𝐷 (𝑓‘𝑢) = (𝐺‘(𝑓 ↾ 𝑢))))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373  ∃wex 1516   ∈ wcel 2177  {cab 2192  ∀wral 2485  ∃wrex 2486  Vcvv 2773   ∪ cun 3168  {csn 3638  ⟨cop 3641  ∪ cuni 3856  Ord word 4417  suc csuc 4420   ↾ cres 4685  Fun wfun 5274  ⟶wf 5276  ‘cfv 5280  recscrecs 6403

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-recs 6404

This theorem is referenced by:  tfrcllemaccex  6460