Theorem tfrlemibfn 6386

Description: The union of 𝐵 is a function defined on 𝑥. Lemma for tfrlemi1 6390. (Contributed by Jim Kingdon, 18-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Hypotheses

Ref	Expression
tfrlemisucfn.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
tfrlemisucfn.2	⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
tfrlemi1.3	⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))}
tfrlemi1.4	⊢ (𝜑 → 𝑥 ∈ On)
tfrlemi1.5	⊢ (𝜑 → ∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))

Assertion

Ref	Expression
tfrlemibfn	⊢ (𝜑 → ∪ 𝐵 Fn 𝑥)

Distinct variable groups:   𝑓,𝑔,ℎ,𝑤,𝑥,𝑦,𝑧,𝐴   𝑓,𝐹,𝑔,ℎ,𝑤,𝑥,𝑦,𝑧   𝜑,𝑤,𝑦   𝑤,𝐵,𝑓,𝑔,ℎ,𝑧   𝜑,𝑔,ℎ,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑓)   𝐵(𝑥,𝑦)

Proof of Theorem tfrlemibfn

Step	Hyp	Ref	Expression
1		tfrlemisucfn.1	. . . . . 6 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2		tfrlemisucfn.2	. . . . . 6 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
3		tfrlemi1.3	. . . . . 6 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))}
4		tfrlemi1.4	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ On)
5		tfrlemi1.5	. . . . . 6 ⊢ (𝜑 → ∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
6	1, 2, 3, 4, 5	tfrlemibacc 6384	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
7	6	unissd 3863	. . . 4 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
8	1	recsfval 6373	. . . 4 ⊢ recs(𝐹) = ∪ 𝐴
9	7, 8	sseqtrrdi 3232	. . 3 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐹))
10	1	tfrlem7 6375	. . 3 ⊢ Fun recs(𝐹)
11		funss 5277	. . 3 ⊢ (∪ 𝐵 ⊆ recs(𝐹) → (Fun recs(𝐹) → Fun ∪ 𝐵))
12	9, 10, 11	mpisyl 1457	. 2 ⊢ (𝜑 → Fun ∪ 𝐵)
13		simpr3 1007	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
14	2	ad2antrr 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
15	4	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑥 ∈ On)
16		simplr 528	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑧 ∈ 𝑥)
17		onelon 4419	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On)
18	15, 16, 17	syl2anc 411	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑧 ∈ On)
19		simpr1 1005	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑔 Fn 𝑧)
20		simpr2 1006	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑔 ∈ 𝐴)
21	1, 14, 18, 19, 20	tfrlemisucfn 6382	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧)
22		dffn2 5409	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧 ↔ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}):suc 𝑧⟶V)
23	21, 22	sylib 122	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}):suc 𝑧⟶V)
24		fssxp 5425	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}):suc 𝑧⟶V → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ⊆ (suc 𝑧 × V))
25	23, 24	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ⊆ (suc 𝑧 × V))
26		eloni 4410	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ On → Ord 𝑥)
27	15, 26	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → Ord 𝑥)
28		ordsucss 4540	. . . . . . . . . . . . . . . 16 ⊢ (Ord 𝑥 → (𝑧 ∈ 𝑥 → suc 𝑧 ⊆ 𝑥))
29	27, 16, 28	sylc 62	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → suc 𝑧 ⊆ 𝑥)
30		xpss1 4773	. . . . . . . . . . . . . . 15 ⊢ (suc 𝑧 ⊆ 𝑥 → (suc 𝑧 × V) ⊆ (𝑥 × V))
31	29, 30	syl 14	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (suc 𝑧 × V) ⊆ (𝑥 × V))
32	25, 31	sstrd 3193	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ⊆ (𝑥 × V))
33		vex 2766	. . . . . . . . . . . . . . . 16 ⊢ 𝑔 ∈ V
34		vex 2766	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
35	2	tfrlem3-2d 6370	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
36	35	simprd 114	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
37		opexg 4261	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
38	34, 36, 37	sylancr 414	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
39		snexg 4217	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
40	38, 39	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
41		unexg 4478	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
42	33, 40, 41	sylancr 414	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
43		elpwg 3613	. . . . . . . . . . . . . . 15 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝒫 (𝑥 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ⊆ (𝑥 × V)))
44	42, 43	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝒫 (𝑥 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ⊆ (𝑥 × V)))
45	44	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝒫 (𝑥 × V) ↔ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ⊆ (𝑥 × V)))
46	32, 45	mpbird 167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝒫 (𝑥 × V))
47	13, 46	eqeltrd 2273	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ ∈ 𝒫 (𝑥 × V))
48	47	ex 115	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑥) → ((𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝑥 × V)))
49	48	exlimdv 1833	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑥) → (∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝑥 × V)))
50	49	rexlimdva 2614	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝑥 × V)))
51	50	abssdv 3257	. . . . . . 7 ⊢ (𝜑 → {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))} ⊆ 𝒫 (𝑥 × V))
52	3, 51	eqsstrid 3229	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝒫 (𝑥 × V))
53		sspwuni 4001	. . . . . 6 ⊢ (𝐵 ⊆ 𝒫 (𝑥 × V) ↔ ∪ 𝐵 ⊆ (𝑥 × V))
54	52, 53	sylib 122	. . . . 5 ⊢ (𝜑 → ∪ 𝐵 ⊆ (𝑥 × V))
55		dmss 4865	. . . . 5 ⊢ (∪ 𝐵 ⊆ (𝑥 × V) → dom ∪ 𝐵 ⊆ dom (𝑥 × V))
56	54, 55	syl 14	. . . 4 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ dom (𝑥 × V))
57		dmxpss 5100	. . . 4 ⊢ dom (𝑥 × V) ⊆ 𝑥
58	56, 57	sstrdi 3195	. . 3 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ 𝑥)
59	1, 2, 3, 4, 5	tfrlemibxssdm 6385	. . 3 ⊢ (𝜑 → 𝑥 ⊆ dom ∪ 𝐵)
60	58, 59	eqssd 3200	. 2 ⊢ (𝜑 → dom ∪ 𝐵 = 𝑥)
61		df-fn 5261	. 2 ⊢ (∪ 𝐵 Fn 𝑥 ↔ (Fun ∪ 𝐵 ∧ dom ∪ 𝐵 = 𝑥))
62	12, 60, 61	sylanbrc 417	1 ⊢ (𝜑 → ∪ 𝐵 Fn 𝑥)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980  ∀wal 1362   = wceq 1364  ∃wex 1506   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∃wrex 2476  Vcvv 2763   ∪ cun 3155   ⊆ wss 3157  𝒫 cpw 3605  {csn 3622  ⟨cop 3625  ∪ cuni 3839  Ord word 4397  Oncon0 4398  suc csuc 4400   × cxp 4661  dom cdm 4663   ↾ cres 4665  Fun wfun 5252   Fn wfn 5253  ⟶wf 5254  ‘cfv 5258  recscrecs 6362

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-recs 6363

This theorem is referenced by:  tfrlemibex  6387  tfrlemiubacc  6388  tfrlemiex  6389