Theorem inf3lema 3460

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 3471 for detailed description.

Hypotheses

Ref	Expression
inf3lem.1	⊢ G = {⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}}
inf3lem.2	⊢ F = (rec(G, ∅) ↾ ω)
inf3lem.3	⊢ A ∈ V
inf3lem.4	⊢ B ∈ V

Assertion

Ref	Expression
inf3lema	⊢ (A ∈ (G ‘B) ↔ (A ∈ x ∧ (A ∩ x) ⊆ B))

Distinct variable group(s):   x,y,z,w

Proof of Theorem inf3lema

Step	Hyp	Ref	Expression
1		inf3lem.1	. . . . . 6 ⊢ G = {⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}}
2		id 9	. . . . . . . 8 ⊢ (z = u → z = u)
3		sseq2 1522	. . . . . . . . . 10 ⊢ (y = v → ((w ∩ x) ⊆ y ↔ (w ∩ x) ⊆ v))
4	3	birabsdv 1344	. . . . . . . . 9 ⊢ (y = v → {w ∈ x∣(w ∩ x) ⊆ y} = {w ∈ x∣(w ∩ x) ⊆ v})
5		ineq1 1638	. . . . . . . . . . 11 ⊢ (w = f → (w ∩ x) = (f ∩ x))
6	5	sseq1d 1527	. . . . . . . . . 10 ⊢ (w = f → ((w ∩ x) ⊆ v ↔ (f ∩ x) ⊆ v))
7	6	cbvrabv 1426	. . . . . . . . 9 ⊢ {w ∈ x∣(w ∩ x) ⊆ v} = {f ∈ x∣(f ∩ x) ⊆ v}
8	4, 7	syl6eq 1140	. . . . . . . 8 ⊢ (y = v → {w ∈ x∣(w ∩ x) ⊆ y} = {f ∈ x∣(f ∩ x) ⊆ v})
9	2, 8	cleqan12rd 1117	. . . . . . 7 ⊢ ((y = v ∧ z = u) → (z = {w ∈ x∣(w ∩ x) ⊆ y} ↔ u = {f ∈ x∣(f ∩ x) ⊆ v}))
10	9	cbvopabv 2105	. . . . . 6 ⊢ {⟨y, z⟩∣z = {w ∈ x∣(w ∩ x) ⊆ y}} = {⟨v, u⟩∣u = {f ∈ x∣(f ∩ x) ⊆ v}}
11	1, 10	eqtr 1119	. . . . 5 ⊢ G = {⟨v, u⟩∣u = {f ∈ x∣(f ∩ x) ⊆ v}}
12	11	fveq1i 2833	. . . 4 ⊢ (G ‘B) = ({⟨v, u⟩∣u = {f ∈ x∣(f ∩ x) ⊆ v}} ‘B)
13		inf3lem.4	. . . . 5 ⊢ B ∈ V
14		visset 1350	. . . . . 6 ⊢ x ∈ V
15	14	rabex 1706	. . . . 5 ⊢ {f ∈ x∣(f ∩ x) ⊆ B} ∈ V
16		sseq2 1522	. . . . . 6 ⊢ (v = B → ((f ∩ x) ⊆ v ↔ (f ∩ x) ⊆ B))
17	16	birabsdv 1344	. . . . 5 ⊢ (v = B → {f ∈ x∣(f ∩ x) ⊆ v} = {f ∈ x∣(f ∩ x) ⊆ B})
18	13, 15, 17	fvopab 2877	. . . 4 ⊢ ({⟨v, u⟩∣u = {f ∈ x∣(f ∩ x) ⊆ v}} ‘B) = {f ∈ x∣(f ∩ x) ⊆ B}
19	12, 18	eqtr 1119	. . 3 ⊢ (G ‘B) = {f ∈ x∣(f ∩ x) ⊆ B}
20	19	eleq2i 1153	. 2 ⊢ (A ∈ (G ‘B) ↔ A ∈ {f ∈ x∣(f ∩ x) ⊆ B})
21		ineq1 1638	. . . 4 ⊢ (f = A → (f ∩ x) = (A ∩ x))
22	21	sseq1d 1527	. . 3 ⊢ (f = A → ((f ∩ x) ⊆ B ↔ (A ∩ x) ⊆ B))
23	22	elrab 1422	. 2 ⊢ (A ∈ {f ∈ x∣(f ∩ x) ⊆ B} ↔ (A ∈ x ∧ (A ∩ x) ⊆ B))
24	20, 23	bitr 151	1 ⊢ (A ∈ (G ‘B) ↔ (A ∈ x ∧ (A ∩ x) ⊆ B))

Syntax hints:   ↔ wb 127   ∧ wa 196   = weq 797   = wceq 1091   ∈ wcel 1092  {crab 1204  Vcvv 1348   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  {copab 2055  ωcom 2372   ↾ cres 2412   ‘cfv 2422  reccrdg 2969

This theorem is referenced by:  inf3lemd 3463  inf3lem1 3464  inf3lem2 3465  inf3lem3 3466

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  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-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

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