Theorem tz9.12lem3 9771

Description: Lemma for tz9.12 9772. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
tz9.12lem.1	⊢ 𝐴 ∈ V
tz9.12lem.2	⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)})

Assertion

Ref	Expression
tz9.12lem3	⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑣,𝐴   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐹(𝑧, 𝑣)

Proof of Theorem tz9.12lem3

Step	Hyp	Ref	Expression
1		tz9.12lem.2	. . . . . . . . . . 11 ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)})
2	1	funmpt2 6567	. . . . . . . . . 10 ⊢ Fun 𝐹
3		fveq2 6873	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑦 → (𝑅1‘𝑣) = (𝑅1‘𝑦))
4	3	eleq2d 2846	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑦 → (𝑥 ∈ (𝑅1‘𝑣) ↔ 𝑥 ∈ (𝑅1‘𝑦)))
5	4	rspcev 3576	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → ∃𝑣 ∈ On 𝑥 ∈ (𝑅1‘𝑣))
6		rabn0 4338	. . . . . . . . . . . . 13 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ On 𝑥 ∈ (𝑅1‘𝑣))
7	5, 6	sylibr 237	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅)
8		intex 5304	. . . . . . . . . . . 12 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
9	7, 8	sylib 221	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
10		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
11		eleq1w 2843	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ (𝑅1‘𝑣) ↔ 𝑥 ∈ (𝑅1‘𝑣)))
12	11	rabbidv 3419	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
13	12	inteqd 4911	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
14	13	eleq1d 2845	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V))
15	1	dmmpt 6230	. . . . . . . . . . . . 13 ⊢ dom 𝐹 = {𝑧 ∈ V ∣ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V}
16	14, 15	elrab2 3648	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ V ∧ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V))
17	10, 16	mpbiran 722	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
18	9, 17	sylibr 237	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → 𝑥 ∈ dom 𝐹)
19		funfvima 7224	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
20	2, 18, 19	sylancr 599	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
21		tz9.12lem.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
22	21, 1	tz9.12lem2 9770	. . . . . . . . . 10 ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On
23	21, 1	tz9.12lem1 9769	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝐴) ⊆ On
24		onsucuni 7822	. . . . . . . . . . . 12 ⊢ ((𝐹 “ 𝐴) ⊆ On → (𝐹 “ 𝐴) ⊆ suc ∪ (𝐹 “ 𝐴))
25	23, 24	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝐴) ⊆ suc ∪ (𝐹 “ 𝐴)
26	25	sseli 3926	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝐹‘𝑥) ∈ suc ∪ (𝐹 “ 𝐴))
27		r1ord2 9763	. . . . . . . . . 10 ⊢ (suc ∪ (𝐹 “ 𝐴) ∈ On → ((𝐹‘𝑥) ∈ suc ∪ (𝐹 “ 𝐴) → (𝑅1‘(𝐹‘𝑥)) ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴))))
28	22, 26, 27	mpsyl 69	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝑅1‘(𝐹‘𝑥)) ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
29	20, 28	syl6 36	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → (𝑥 ∈ 𝐴 → (𝑅1‘(𝐹‘𝑥)) ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴))))
30	29	imp 412	. . . . . . 7 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝑅1‘(𝐹‘𝑥)) ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
31	13, 1	fvmptg 6979	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V) → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
32	10, 31	mpan 703	. . . . . . . . . . 11 ⊢ (∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
33	8, 32	sylbi 220	. . . . . . . . . 10 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
34		ssrab2 4027	. . . . . . . . . . 11 ⊢ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ⊆ On
35		onint 7787	. . . . . . . . . . 11 ⊢ (({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ⊆ On ∧ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅) → ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
36	34, 35	mpan 703	. . . . . . . . . 10 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ → ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
37	33, 36	eqeltrd 2860	. . . . . . . . 9 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ → (𝐹‘𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
38		fveq2 6873	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑅1‘𝑦) = (𝑅1‘(𝐹‘𝑥)))
39	38	eleq2d 2846	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥 ∈ (𝑅1‘𝑦) ↔ 𝑥 ∈ (𝑅1‘(𝐹‘𝑥))))
40	4	cbvrabv 3422	. . . . . . . . . . 11 ⊢ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑦)}
41	39, 40	elrab2 3648	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ↔ ((𝐹‘𝑥) ∈ On ∧ 𝑥 ∈ (𝑅1‘(𝐹‘𝑥))))
42	41	simprbi 503	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ (𝑅1‘(𝐹‘𝑥)))
43	7, 37, 42	3syl 19	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → 𝑥 ∈ (𝑅1‘(𝐹‘𝑥)))
44	43	adantr 486	. . . . . . 7 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑅1‘(𝐹‘𝑥)))
45	30, 44	sseldd 3931	. . . . . 6 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
46	45	exp31 425	. . . . 5 ⊢ (𝑦 ∈ On → (𝑥 ∈ (𝑅1‘𝑦) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))))
47	46	com3r 88	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ On → (𝑥 ∈ (𝑅1‘𝑦) → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))))
48	47	rexlimdv 3161	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴))))
49	48	ralimia 3096	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
50		r1suc 9752	. . . . 5 ⊢ (suc ∪ (𝐹 “ 𝐴) ∈ On → (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)) = 𝒫 (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
51	22, 50	ax-mp 5	. . . 4 ⊢ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)) = 𝒫 (𝑅1‘suc ∪ (𝐹 “ 𝐴))
52	51	eleq2i 2852	. . 3 ⊢ (𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)) ↔ 𝐴 ∈ 𝒫 (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
53	21	elpw 4560	. . 3 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘suc ∪ (𝐹 “ 𝐴)) ↔ 𝐴 ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
54		dfss3 3919	. . 3 ⊢ (𝐴 ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
55	52, 53, 54	3bitri 300	. 2 ⊢ (𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
56	49, 55	sylibr 237	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3898  ∅c0 4278  𝒫 cpw 4556  ∪ cuni 4866  ∩ cint 4906   ↦ cmpt 5185  dom cdm 5647   “ cima 5650  Oncon0 6351  suc csuc 6353  Fun wfun 6521  ‘cfv 6527  𝑅₁cr1 9744

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-r1 9746

This theorem is used by:  tz9.12  9772