Theorem tfrlemibfn 6233

Description: The union of 𝐵 is a function defined on 𝑥. Lemma for tfrlemi1 6237. (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 6231	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
7	6	unissd 3768	. . . 4 ⊢ (𝜑 → ∪ 𝐵 ⊆ ∪ 𝐴)
8	1	recsfval 6220	. . . 4 ⊢ recs(𝐹) = ∪ 𝐴
9	7, 8	sseqtrrdi 3151	. . 3 ⊢ (𝜑 → ∪ 𝐵 ⊆ recs(𝐹))
10	1	tfrlem7 6222	. . 3 ⊢ Fun recs(𝐹)
11		funss 5150	. . 3 ⊢ (∪ 𝐵 ⊆ recs(𝐹) → (Fun recs(𝐹) → Fun ∪ 𝐵))
12	9, 10, 11	mpisyl 1423	. 2 ⊢ (𝜑 → Fun ∪ 𝐵)
13		simpr3 990	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
14	2	ad2antrr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
15	4	ad2antrr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑥 ∈ On)
16		simplr 520	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑧 ∈ 𝑥)
17		onelon 4314	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On)
18	15, 16, 17	syl2anc 409	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑧 ∈ On)
19		simpr1 988	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑔 Fn 𝑧)
20		simpr2 989	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → 𝑔 ∈ 𝐴)
21	1, 14, 18, 19, 20	tfrlemisucfn 6229	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧)
22		dffn2 5282	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) Fn suc 𝑧 ↔ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}):suc 𝑧⟶V)
23	21, 22	sylib 121	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}):suc 𝑧⟶V)
24		fssxp 5298	. . . . . . . . . . . . . . 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 4305	. . . . . . . . . . . . . . . . 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 4428	. . . . . . . . . . . . . . . 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 4657	. . . . . . . . . . . . . . 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 3112	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ⊆ (𝑥 × V))
33		vex 2692	. . . . . . . . . . . . . . . 16 ⊢ 𝑔 ∈ V
34		vex 2692	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
35	2	tfrlem3-2d 6217	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
36	35	simprd 113	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
37		opexg 4158	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
38	34, 36, 37	sylancr 411	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
39		snexg 4116	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
40	38, 39	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V)
41		unexg 4372	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹‘𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
42	33, 40, 41	sylancr 411	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ V)
43		elpwg 3523	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . 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 166	. . . . . . . . . . . 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 2217	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))) → ℎ ∈ 𝒫 (𝑥 × V))
48	47	ex 114	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑥) → ((𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝑥 × V)))
49	48	exlimdv 1792	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑥) → (∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝑥 × V)))
50	49	rexlimdva 2552	. . . . . . . 8 ⊢ (𝜑 → (∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ℎ ∈ 𝒫 (𝑥 × V)))
51	50	abssdv 3176	. . . . . . 7 ⊢ (𝜑 → {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))} ⊆ 𝒫 (𝑥 × V))
52	3, 51	eqsstrid 3148	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝒫 (𝑥 × V))
53		sspwuni 3905	. . . . . 6 ⊢ (𝐵 ⊆ 𝒫 (𝑥 × V) ↔ ∪ 𝐵 ⊆ (𝑥 × V))
54	52, 53	sylib 121	. . . . 5 ⊢ (𝜑 → ∪ 𝐵 ⊆ (𝑥 × V))
55		dmss 4746	. . . . 5 ⊢ (∪ 𝐵 ⊆ (𝑥 × V) → dom ∪ 𝐵 ⊆ dom (𝑥 × V))
56	54, 55	syl 14	. . . 4 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ dom (𝑥 × V))
57		dmxpss 4977	. . . 4 ⊢ dom (𝑥 × V) ⊆ 𝑥
58	56, 57	sstrdi 3114	. . 3 ⊢ (𝜑 → dom ∪ 𝐵 ⊆ 𝑥)
59	1, 2, 3, 4, 5	tfrlemibxssdm 6232	. . 3 ⊢ (𝜑 → 𝑥 ⊆ dom ∪ 𝐵)
60	58, 59	eqssd 3119	. 2 ⊢ (𝜑 → dom ∪ 𝐵 = 𝑥)
61		df-fn 5134	. 2 ⊢ (∪ 𝐵 Fn 𝑥 ↔ (Fun ∪ 𝐵 ∧ dom ∪ 𝐵 = 𝑥))
62	12, 60, 61	sylanbrc 414	1 ⊢ (𝜑 → ∪ 𝐵 Fn 𝑥)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 963  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∃wrex 2418  Vcvv 2689   ∪ cun 3074   ⊆ wss 3076  𝒫 cpw 3515  {csn 3532  ⟨cop 3535  ∪ cuni 3744  Ord word 4292  Oncon0 4293  suc csuc 4295   × cxp 4545  dom cdm 4547   ↾ cres 4549  Fun wfun 5125   Fn wfn 5126  ⟶wf 5127  ‘cfv 5131  recscrecs 6209

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-fv 5139  df-recs 6210

This theorem is referenced by:  tfrlemibex  6234  tfrlemiubacc  6235  tfrlemiex  6236