Theorem tfrlemibfn 6603

Description: The union of 𝐵 is a function defined on 𝑥. Lemma for tfrlemi1 6607. (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 6601	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
7	6	unissd 3959	. . . 4 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
8	1	recsfval 6590	. . . 4 ⊢ recs(𝐹) = ∪ 𝐴
9	7, 8	sseqtrrdi 3297	. . 3 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐹))
10	1	tfrlem7 6592	. . 3 ⊢ Fun recs(𝐹)
11		funss 5399	. . 3 ⊢ (∪ 𝐵 ⊆ recs(𝐹) → (Fun recs(𝐹) → Fun ∪ 𝐵))
12	9, 10, 11	mpisyl 1496	. 2 ⊢ (𝜑 → Fun ∪ 𝐵)
13		simpr3 1036	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
14	2	ad2antrr 492	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
15	4	ad2antrr 492	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑥 ∈ On)
16		simplr 533	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑧 ∈ 𝑥)
17		onelon 4531	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On)
18	15, 16, 17	syl2anc 415	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑧 ∈ On)
19		simpr1 1034	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑔 Fn 𝑧)
20		simpr2 1035	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑔 ∈ 𝐴)
21	1, 14, 18, 19, 20	tfrlemisucfn 6599	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧)
22		dffn2 5538	. . . . . . . . . . . . . . . 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 5558	. . . . . . . . . . . . . . 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 4522	. . . . . . . . . . . . . . . . 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 4653	. . . . . . . . . . . . . . . 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 4887	. . . . . . . . . . . . . . 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 3258	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ⊆ (𝑥 × V))
33		vex 2824	. . . . . . . . . . . . . . . 16 ⊢ 𝑔 ∈ V
34		vex 2824	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
35	2	tfrlem3-2d 6587	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
36	35	simprd 114	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
37		opexg 4370	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
38	34, 36, 37	sylancr 418	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
39		snexg 4323	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
40	38, 39	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
41		unexg 4591	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
42	33, 40, 41	sylancr 418	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
43		elpwg 3696	. . . . . . . . . . . . . . 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 492	. . . . . . . . . . . . 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 2315	. . . . . . . . . . 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 1872	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑥) → (∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝑥 × V)))
50	49	rexlimdva 2668	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝑥 × V)))
51	50	abssdv 3322	. . . . . . 7 ⊢ (𝜑 → {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))} ⊆ 𝒫 (𝑥 × V))
52	3, 51	eqsstrid 3294	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝒫 (𝑥 × V))
53		sspwuni 4097	. . . . . 6 ⊢ (𝐵 ⊆ 𝒫 (𝑥 × V) ↔ ∪ 𝐵 ⊆ (𝑥 × V))
54	52, 53	sylib 122	. . . . 5 ⊢ (𝜑 → ∪ 𝐵 ⊆ (𝑥 × V))
55		dmss 4982	. . . . 5 ⊢ (∪ 𝐵 ⊆ (𝑥 × V) → dom ∪ 𝐵 ⊆ dom (𝑥 × V))
56	54, 55	syl 14	. . . 4 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ dom (𝑥 × V))
57		dmxpss 5221	. . . 4 ⊢ dom (𝑥 × V) ⊆ 𝑥
58	56, 57	sstrdi 3260	. . 3 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ 𝑥)
59	1, 2, 3, 4, 5	tfrlemibxssdm 6602	. . 3 ⊢ (𝜑 → 𝑥 ⊆ dom ∪ 𝐵)
60	58, 59	eqssd 3265	. 2 ⊢ (𝜑 → dom ∪ 𝐵 = 𝑥)
61		df-fn 5383	. 2 ⊢ (∪ 𝐵 Fn 𝑥 ↔ (Fun ∪ 𝐵 ∧ dom ∪ 𝐵 = 𝑥))
62	12, 60, 61	sylanbrc 421	1 ⊢ (𝜑 → ∪ 𝐵 Fn 𝑥)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ∪ cun 3218   ⊆ wss 3220  𝒫 cpw 3688  {csn 3709  ⟨cop 3712  ∪ cuni 3935  Ord word 4509  Oncon0 4510  suc csuc 4512   × cxp 4774  dom cdm 4776   ↾ cres 4778  Fun wfun 5374   Fn wfn 5375  ⟶wf 5376  ‘cfv 5380  recscrecs 6579

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-recs 6580

This theorem is used by:  tfrlemibex  6604  tfrlemiubacc  6605  tfrlemiex  6606