Theorem tz9.12lem3 9705

Description: Lemma for tz9.12 9706. (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 6532	. . . . . . . . . 10 ⊢ Fun 𝐹
3		fveq2 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑦 → (𝑅1‘𝑣) = (𝑅1‘𝑦))
4	3	eleq2d 2823	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑦 → (𝑥 ∈ (𝑅1‘𝑣) ↔ 𝑥 ∈ (𝑅1‘𝑦)))
5	4	rspcev 3577	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → ∃𝑣 ∈ On 𝑥 ∈ (𝑅1‘𝑣))
6		rabn0 4342	. . . . . . . . . . . . 13 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ On 𝑥 ∈ (𝑅1‘𝑣))
7	5, 6	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅)
8		intex 5290	. . . . . . . . . . . 12 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
9	7, 8	sylib 218	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
10		vex 3445	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
11		eleq1w 2820	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ (𝑅1‘𝑣) ↔ 𝑥 ∈ (𝑅1‘𝑣)))
12	11	rabbidv 3407	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
13	12	inteqd 4908	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
14	13	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V))
15	1	dmmpt 6199	. . . . . . . . . . . . 13 ⊢ dom 𝐹 = {𝑧 ∈ V ∣ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V}
16	14, 15	elrab2 3650	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ V ∧ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V))
17	10, 16	mpbiran 710	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
18	9, 17	sylibr 234	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → 𝑥 ∈ dom 𝐹)
19		funfvima 7178	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
20	2, 18, 19	sylancr 588	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
21		tz9.12lem.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
22	21, 1	tz9.12lem2 9704	. . . . . . . . . 10 ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On
23	21, 1	tz9.12lem1 9703	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝐴) ⊆ On
24		onsucuni 7772	. . . . . . . . . . . 12 ⊢ ((𝐹 “ 𝐴) ⊆ On → (𝐹 “ 𝐴) ⊆ suc ∪ (𝐹 “ 𝐴))
25	23, 24	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝐴) ⊆ suc ∪ (𝐹 “ 𝐴)
26	25	sseli 3930	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝐹‘𝑥) ∈ suc ∪ (𝐹 “ 𝐴))
27		r1ord2 9697	. . . . . . . . . 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 406	. . . . . . 7 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝑅1‘(𝐹‘𝑥)) ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
31	13, 1	fvmptg 6940	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V) → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
32	10, 31	mpan 691	. . . . . . . . . . 11 ⊢ (∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
33	8, 32	sylbi 217	. . . . . . . . . 10 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
34		ssrab2 4033	. . . . . . . . . . 11 ⊢ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ⊆ On
35		onint 7737	. . . . . . . . . . 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 691	. . . . . . . . . 10 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ → ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
37	33, 36	eqeltrd 2837	. . . . . . . . 9 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ → (𝐹‘𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
38		fveq2 6835	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑅1‘𝑦) = (𝑅1‘(𝐹‘𝑥)))
39	38	eleq2d 2823	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥 ∈ (𝑅1‘𝑦) ↔ 𝑥 ∈ (𝑅1‘(𝐹‘𝑥))))
40	4	cbvrabv 3410	. . . . . . . . . . 11 ⊢ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑦)}
41	39, 40	elrab2 3650	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ↔ ((𝐹‘𝑥) ∈ On ∧ 𝑥 ∈ (𝑅1‘(𝐹‘𝑥))))
42	41	simprbi 496	. . . . . . . . 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 480	. . . . . . 7 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑅1‘(𝐹‘𝑥)))
45	30, 44	sseldd 3935	. . . . . 6 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
46	45	exp31 419	. . . . 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 3136	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴))))
49	48	ralimia 3071	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
50		r1suc 9686	. . . . 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 2829	. . 3 ⊢ (𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)) ↔ 𝐴 ∈ 𝒫 (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
53	21	elpw 4559	. . 3 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘suc ∪ (𝐹 “ 𝐴)) ↔ 𝐴 ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
54		dfss3 3923	. . 3 ⊢ (𝐴 ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
55	52, 53, 54	3bitri 297	. 2 ⊢ (𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
56	49, 55	sylibr 234	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3400  Vcvv 3441   ⊆ wss 3902  ∅c0 4286  𝒫 cpw 4555  ∪ cuni 4864  ∩ cint 4903   ↦ cmpt 5180  dom cdm 5625   “ cima 5628  Oncon0 6318  suc csuc 6320  Fun wfun 6487  ‘cfv 6493  𝑅₁cr1 9678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-om 7811  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-r1 9680

This theorem is referenced by:  tz9.12  9706