Theorem tz9.12lem3 9010

Description: Lemma for tz9.12 9011. (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 6224	. . . . . . . . . 10 ⊢ Fun 𝐹
3		fveq2 6496	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑦 → (𝑅1‘𝑣) = (𝑅1‘𝑦))
4	3	eleq2d 2845	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑦 → (𝑥 ∈ (𝑅1‘𝑣) ↔ 𝑥 ∈ (𝑅1‘𝑦)))
5	4	rspcev 3529	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → ∃𝑣 ∈ On 𝑥 ∈ (𝑅1‘𝑣))
6		rabn0 4219	. . . . . . . . . . . . 13 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ On 𝑥 ∈ (𝑅1‘𝑣))
7	5, 6	sylibr 226	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅)
8		intex 5092	. . . . . . . . . . . 12 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
9	7, 8	sylib 210	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
10		vex 3412	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
11		eleq1w 2842	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ (𝑅1‘𝑣) ↔ 𝑥 ∈ (𝑅1‘𝑣)))
12	11	rabbidv 3397	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
13	12	inteqd 4750	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
14	13	eleq1d 2844	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V))
15	1	dmmpt 5930	. . . . . . . . . . . . 13 ⊢ dom 𝐹 = {𝑧 ∈ V ∣ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V}
16	14, 15	elrab2 3593	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ V ∧ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V))
17	10, 16	mpbiran 696	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
18	9, 17	sylibr 226	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → 𝑥 ∈ dom 𝐹)
19		funfvima 6816	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
20	2, 18, 19	sylancr 578	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
21		tz9.12lem.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
22	21, 1	tz9.12lem2 9009	. . . . . . . . . 10 ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On
23	21, 1	tz9.12lem1 9008	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝐴) ⊆ On
24		onsucuni 7357	. . . . . . . . . . . 12 ⊢ ((𝐹 “ 𝐴) ⊆ On → (𝐹 “ 𝐴) ⊆ suc ∪ (𝐹 “ 𝐴))
25	23, 24	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝐴) ⊆ suc ∪ (𝐹 “ 𝐴)
26	25	sseli 3848	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝐹‘𝑥) ∈ suc ∪ (𝐹 “ 𝐴))
27		r1ord2 9002	. . . . . . . . . 10 ⊢ (suc ∪ (𝐹 “ 𝐴) ∈ On → ((𝐹‘𝑥) ∈ suc ∪ (𝐹 “ 𝐴) → (𝑅1‘(𝐹‘𝑥)) ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴))))
28	22, 26, 27	mpsyl 68	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝑅1‘(𝐹‘𝑥)) ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
29	20, 28	syl6 35	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → (𝑥 ∈ 𝐴 → (𝑅1‘(𝐹‘𝑥)) ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴))))
30	29	imp 398	. . . . . . 7 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝑅1‘(𝐹‘𝑥)) ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
31	13, 1	fvmptg 6591	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V) → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
32	10, 31	mpan 677	. . . . . . . . . . 11 ⊢ (∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
33	8, 32	sylbi 209	. . . . . . . . . 10 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
34		ssrab2 3940	. . . . . . . . . . 11 ⊢ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ⊆ On
35		onint 7324	. . . . . . . . . . 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 677	. . . . . . . . . 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 6496	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑅1‘𝑦) = (𝑅1‘(𝐹‘𝑥)))
39	38	eleq2d 2845	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥 ∈ (𝑅1‘𝑦) ↔ 𝑥 ∈ (𝑅1‘(𝐹‘𝑥))))
40	4	cbvrabv 3406	. . . . . . . . . . 11 ⊢ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑦)}
41	39, 40	elrab2 3593	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ↔ ((𝐹‘𝑥) ∈ On ∧ 𝑥 ∈ (𝑅1‘(𝐹‘𝑥))))
42	41	simprbi 489	. . . . . . . . 9 ⊢ ((𝐹‘𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} → 𝑥 ∈ (𝑅1‘(𝐹‘𝑥)))
43	7, 37, 42	3syl 18	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → 𝑥 ∈ (𝑅1‘(𝐹‘𝑥)))
44	43	adantr 473	. . . . . . 7 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑅1‘(𝐹‘𝑥)))
45	30, 44	sseldd 3853	. . . . . 6 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
46	45	exp31 412	. . . . 5 ⊢ (𝑦 ∈ On → (𝑥 ∈ (𝑅1‘𝑦) → (𝑥 ∈ 𝐴 → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))))
47	46	com3r 87	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑦 ∈ On → (𝑥 ∈ (𝑅1‘𝑦) → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))))
48	47	rexlimdv 3222	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴))))
49	48	ralimia 3102	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
50		r1suc 8991	. . . . 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 2851	. . 3 ⊢ (𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)) ↔ 𝐴 ∈ 𝒫 (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
53	21	elpw 4422	. . 3 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘suc ∪ (𝐹 “ 𝐴)) ↔ 𝐴 ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
54		dfss3 3841	. . 3 ⊢ (𝐴 ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
55	52, 53, 54	3bitri 289	. 2 ⊢ (𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
56	49, 55	sylibr 226	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050   ≠ wne 2961  ∀wral 3082  ∃wrex 3083  {crab 3086  Vcvv 3409   ⊆ wss 3823  ∅c0 4172  𝒫 cpw 4416  ∪ cuni 4708  ∩ cint 4745   ↦ cmpt 5004  dom cdm 5403   “ cima 5406  Oncon0 6026  suc csuc 6028  Fun wfun 6179  ‘cfv 6185  𝑅₁cr1 8983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-om 7395  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-r1 8985

This theorem is referenced by:  tz9.12  9011