Theorem tz9.12lem3 8816

Description: Lemma for tz9.12 8817. (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 6070	. . . . . . . . . 10 ⊢ Fun 𝐹
3		fveq2 6332	. . . . . . . . . . . . . . 15 ⊢ (𝑣 = 𝑦 → (𝑅1‘𝑣) = (𝑅1‘𝑦))
4	3	eleq2d 2836	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑦 → (𝑥 ∈ (𝑅1‘𝑣) ↔ 𝑥 ∈ (𝑅1‘𝑦)))
5	4	rspcev 3460	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → ∃𝑣 ∈ On 𝑥 ∈ (𝑅1‘𝑣))
6		rabn0 4104	. . . . . . . . . . . . 13 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∃𝑣 ∈ On 𝑥 ∈ (𝑅1‘𝑣))
7	5, 6	sylibr 224	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅)
8		intex 4951	. . . . . . . . . . . 12 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
9	7, 8	sylib 208	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
10		vex 3354	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
11		eleq1w 2833	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ (𝑅1‘𝑣) ↔ 𝑥 ∈ (𝑅1‘𝑣)))
12	11	rabbidv 3339	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} = {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
13	12	inteqd 4616	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
14	13	eleq1d 2835	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V))
15	1	dmmpt 5774	. . . . . . . . . . . . 13 ⊢ dom 𝐹 = {𝑧 ∈ V ∣ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)} ∈ V}
16	14, 15	elrab2 3518	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ dom 𝐹 ↔ (𝑥 ∈ V ∧ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V))
17	10, 16	mpbiran 688	. . . . . . . . . . 11 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V)
18	9, 17	sylibr 224	. . . . . . . . . 10 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → 𝑥 ∈ dom 𝐹)
19		funfvima 6635	. . . . . . . . . 10 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
20	2, 18, 19	sylancr 575	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴)))
21		tz9.12lem.1	. . . . . . . . . . 11 ⊢ 𝐴 ∈ V
22	21, 1	tz9.12lem2 8815	. . . . . . . . . 10 ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On
23	21, 1	tz9.12lem1 8814	. . . . . . . . . . . 12 ⊢ (𝐹 “ 𝐴) ⊆ On
24		onsucuni 7175	. . . . . . . . . . . 12 ⊢ ((𝐹 “ 𝐴) ⊆ On → (𝐹 “ 𝐴) ⊆ suc ∪ (𝐹 “ 𝐴))
25	23, 24	ax-mp 5	. . . . . . . . . . 11 ⊢ (𝐹 “ 𝐴) ⊆ suc ∪ (𝐹 “ 𝐴)
26	25	sseli 3748	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝐹‘𝑥) ∈ suc ∪ (𝐹 “ 𝐴))
27		r1ord2 8808	. . . . . . . . . 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 393	. . . . . . 7 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → (𝑅1‘(𝐹‘𝑥)) ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
31	13, 1	fvmptg 6422	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V) → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
32	10, 31	mpan 670	. . . . . . . . . . 11 ⊢ (∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ V → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
33	8, 32	sylbi 207	. . . . . . . . . 10 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ → (𝐹‘𝑥) = ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
34		ssrab2 3836	. . . . . . . . . . 11 ⊢ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ⊆ On
35		onint 7142	. . . . . . . . . . 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 670	. . . . . . . . . 10 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ → ∩ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
37	33, 36	eqeltrd 2850	. . . . . . . . 9 ⊢ ({𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ≠ ∅ → (𝐹‘𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)})
38		fveq2 6332	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑅1‘𝑦) = (𝑅1‘(𝐹‘𝑥)))
39	38	eleq2d 2836	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑥 ∈ (𝑅1‘𝑦) ↔ 𝑥 ∈ (𝑅1‘(𝐹‘𝑥))))
40	4	cbvrabv 3349	. . . . . . . . . . 11 ⊢ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} = {𝑦 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑦)}
41	39, 40	elrab2 3518	. . . . . . . . . 10 ⊢ ((𝐹‘𝑥) ∈ {𝑣 ∈ On ∣ 𝑥 ∈ (𝑅1‘𝑣)} ↔ ((𝐹‘𝑥) ∈ On ∧ 𝑥 ∈ (𝑅1‘(𝐹‘𝑥))))
42	41	simprbi 484	. . . . . . . . 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 466	. . . . . . 7 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑅1‘(𝐹‘𝑥)))
45	30, 44	sseldd 3753	. . . . . 6 ⊢ (((𝑦 ∈ On ∧ 𝑥 ∈ (𝑅1‘𝑦)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
46	45	exp31 406	. . . . 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 3178	. . 3 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴))))
49	48	ralimia 3099	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
50		r1suc 8797	. . . . 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 2842	. . 3 ⊢ (𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)) ↔ 𝐴 ∈ 𝒫 (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
53	21	elpw 4303	. . 3 ⊢ (𝐴 ∈ 𝒫 (𝑅1‘suc ∪ (𝐹 “ 𝐴)) ↔ 𝐴 ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
54		dfss3 3741	. . 3 ⊢ (𝐴 ⊆ (𝑅1‘suc ∪ (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
55	52, 53, 54	3bitri 286	. 2 ⊢ (𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ (𝑅1‘suc ∪ (𝐹 “ 𝐴)))
56	49, 55	sylibr 224	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ On 𝑥 ∈ (𝑅1‘𝑦) → 𝐴 ∈ (𝑅1‘suc suc ∪ (𝐹 “ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  ∀wral 3061  ∃wrex 3062  {crab 3065  Vcvv 3351   ⊆ wss 3723  ∅c0 4063  𝒫 cpw 4297  ∪ cuni 4574  ∩ cint 4611   ↦ cmpt 4863  dom cdm 5249   “ cima 5252  Oncon0 5866  suc csuc 5868  Fun wfun 6025  ‘cfv 6031  𝑅₁cr1 8789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-r1 8791

This theorem is referenced by:  tz9.12  8817