Theorem tz9.12lem1 3503

Description: Lemma for tz9.12 3506.

Hypotheses

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

Assertion

Ref	Expression
tz9.12lem1	⊢ (F “ A) ⊆ On

Distinct variable group(s):   z,w,v,A

Proof of Theorem tz9.12lem1

Step	Hyp	Ref	Expression
1		visset 1350	. . . 4 ⊢ y ∈ V
2	1	elima3 2608	. . 3 ⊢ (y ∈ (F “ A) ↔ ∃x(x ∈ A ∧ ⟨x, y⟩ ∈ F))
3		tz9.12lem.2	. . . . . . . 8 ⊢ F = {⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}}
4	3	eleq2i 1153	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ F ↔ ⟨x, y⟩ ∈ {⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}})
5		visset 1350	. . . . . . . 8 ⊢ x ∈ V
6		eleq1 1149	. . . . . . . . . . 11 ⊢ (z = x → (z ∈ (R1 ‘v) ↔ x ∈ (R1 ‘v)))
7	6	birabsdv 1344	. . . . . . . . . 10 ⊢ (z = x → {v ∈ On∣z ∈ (R1 ‘v)} = {v ∈ On∣x ∈ (R1 ‘v)})
8	7	inteqd 1970	. . . . . . . . 9 ⊢ (z = x → ∩{v ∈ On∣z ∈ (R1 ‘v)} = ∩{v ∈ On∣x ∈ (R1 ‘v)})
9	8	cleq2d 1112	. . . . . . . 8 ⊢ (z = x → (w = ∩{v ∈ On∣z ∈ (R1 ‘v)} ↔ w = ∩{v ∈ On∣x ∈ (R1 ‘v)}))
10		cleq1 1107	. . . . . . . 8 ⊢ (w = y → (w = ∩{v ∈ On∣x ∈ (R1 ‘v)} ↔ y = ∩{v ∈ On∣x ∈ (R1 ‘v)}))
11	5, 1, 9, 10	opelopab 2117	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ {⟨z, w⟩∣w = ∩{v ∈ On∣z ∈ (R1 ‘v)}} ↔ y = ∩{v ∈ On∣x ∈ (R1 ‘v)})
12	4, 11	bitr 151	. . . . . 6 ⊢ (⟨x, y⟩ ∈ F ↔ y = ∩{v ∈ On∣x ∈ (R1 ‘v)})
13		19.8a 712	. . . . . . . 8 ⊢ (y = ∩{v ∈ On∣x ∈ (R1 ‘v)} → ∃y y = ∩{v ∈ On∣x ∈ (R1 ‘v)})
14		isset 1351	. . . . . . . 8 ⊢ (∩{v ∈ On∣x ∈ (R1 ‘v)} ∈ V ↔ ∃y y = ∩{v ∈ On∣x ∈ (R1 ‘v)})
15	13, 14	sylibr 175	. . . . . . 7 ⊢ (y = ∩{v ∈ On∣x ∈ (R1 ‘v)} → ∩{v ∈ On∣x ∈ (R1 ‘v)} ∈ V)
16		intex 1986	. . . . . . . 8 ⊢ (¬ {v ∈ On∣x ∈ (R1 ‘v)} = ∅ ↔ ∩{v ∈ On∣x ∈ (R1 ‘v)} ∈ V)
17		eleq1 1149	. . . . . . . . . 10 ⊢ (y = ∩{v ∈ On∣x ∈ (R1 ‘v)} → (y ∈ On ↔ ∩{v ∈ On∣x ∈ (R1 ‘v)} ∈ On))
18		ssrab 1556	. . . . . . . . . . 11 ⊢ {v ∈ On∣x ∈ (R1 ‘v)} ⊆ On
19		oninton 2267	. . . . . . . . . . 11 ⊢ (({v ∈ On∣x ∈ (R1 ‘v)} ⊆ On ∧ ¬ {v ∈ On∣x ∈ (R1 ‘v)} = ∅) → ∩{v ∈ On∣x ∈ (R1 ‘v)} ∈ On)
20	18, 19	mpan 518	. . . . . . . . . 10 ⊢ (¬ {v ∈ On∣x ∈ (R1 ‘v)} = ∅ → ∩{v ∈ On∣x ∈ (R1 ‘v)} ∈ On)
21	17, 20	syl5bir 184	. . . . . . . . 9 ⊢ (y = ∩{v ∈ On∣x ∈ (R1 ‘v)} → (¬ {v ∈ On∣x ∈ (R1 ‘v)} = ∅ → y ∈ On))
22	21	com12 13	. . . . . . . 8 ⊢ (¬ {v ∈ On∣x ∈ (R1 ‘v)} = ∅ → (y = ∩{v ∈ On∣x ∈ (R1 ‘v)} → y ∈ On))
23	16, 22	sylbir 176	. . . . . . 7 ⊢ (∩{v ∈ On∣x ∈ (R1 ‘v)} ∈ V → (y = ∩{v ∈ On∣x ∈ (R1 ‘v)} → y ∈ On))
24	15, 23	mpcom 49	. . . . . 6 ⊢ (y = ∩{v ∈ On∣x ∈ (R1 ‘v)} → y ∈ On)
25	12, 24	sylbi 174	. . . . 5 ⊢ (⟨x, y⟩ ∈ F → y ∈ On)
26	25	adantl 305	. . . 4 ⊢ ((x ∈ A ∧ ⟨x, y⟩ ∈ F) → y ∈ On)
27	26	19.23aiv 952	. . 3 ⊢ (∃x(x ∈ A ∧ ⟨x, y⟩ ∈ F) → y ∈ On)
28	2, 27	sylbi 174	. 2 ⊢ (y ∈ (F “ A) → y ∈ On)
29	28	ssriv 1508	1 ⊢ (F “ A) ⊆ On

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ⟨cop 1810  ∩cint 1965  {copab 2055  Oncon0 2199   “ cima 2413   ‘cfv 2422  R₁cr1 3485

This theorem is referenced by:  tz9.12lem2 3504  tz9.12lem3 3505

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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