Theorem inf3lema 9607

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9618 for detailed description. (Contributed by NM, 28-Oct-1996.)

Hypotheses

Ref	Expression
inf3lem.1	⊢ 𝐺 = (𝑦 ∈ V ↦ {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:   𝑥,𝑦,𝑤

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑤)   𝐵(𝑥, 𝑦, 𝑤)   𝐹(𝑥, 𝑦, 𝑤)   𝐺(𝑥, 𝑦, 𝑤)

Proof of Theorem inf3lema

Dummy variables 𝑣 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 4162	. . 3 ⊢ (𝑓 = 𝐴 → (𝑓 ∩ 𝑥) = (𝐴 ∩ 𝑥))
2	1	sseq1d 3965	. 2 ⊢ (𝑓 = 𝐴 → ((𝑓 ∩ 𝑥) ⊆ 𝐵 ↔ (𝐴 ∩ 𝑥) ⊆ 𝐵))
3		inf3lem.4	. . 3 ⊢ 𝐵 ∈ V
4		sseq2 3960	. . . . 5 ⊢ (𝑣 = 𝐵 → ((𝑓 ∩ 𝑥) ⊆ 𝑣 ↔ (𝑓 ∩ 𝑥) ⊆ 𝐵))
5	4	rabbidv 3421	. . . 4 ⊢ (𝑣 = 𝐵 → {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣} = {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝐵})
6		inf3lem.1	. . . . 5 ⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
7		sseq2 3960	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑤 ∩ 𝑥) ⊆ 𝑦 ↔ (𝑤 ∩ 𝑥) ⊆ 𝑣))
8	7	rabbidv 3421	. . . . . . 7 ⊢ (𝑦 = 𝑣 → {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦} = {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑣})
9		ineq1 4162	. . . . . . . . 9 ⊢ (𝑤 = 𝑓 → (𝑤 ∩ 𝑥) = (𝑓 ∩ 𝑥))
10	9	sseq1d 3965	. . . . . . . 8 ⊢ (𝑤 = 𝑓 → ((𝑤 ∩ 𝑥) ⊆ 𝑣 ↔ (𝑓 ∩ 𝑥) ⊆ 𝑣))
11	10	cbvrabv 3424	. . . . . . 7 ⊢ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑣} = {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣}
12	8, 11	eqtrdi 2813	. . . . . 6 ⊢ (𝑦 = 𝑣 → {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦} = {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
13	12	cbvmptv 5213	. . . . 5 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑣 ∈ V ↦ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
14	6, 13	eqtri 2785	. . . 4 ⊢ 𝐺 = (𝑣 ∈ V ↦ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
15		vex 3457	. . . . 5 ⊢ 𝑥 ∈ V
16	15	rabex 5307	. . . 4 ⊢ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝐵} ∈ V
17	5, 14, 16	fvmpt 6990	. . 3 ⊢ (𝐵 ∈ V → (𝐺‘𝐵) = {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝐵})
18	3, 17	ax-mp 5	. 2 ⊢ (𝐺‘𝐵) = {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝐵}
19	2, 18	elrab2 3652	1 ⊢ (𝐴 ∈ (𝐺‘𝐵) ↔ (𝐴 ∈ 𝑥 ∧ (𝐴 ∩ 𝑥) ⊆ 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282   ↦ cmpt 5190   ↾ cres 5661  ‘cfv 6537  ωcom 7866  reccrdg 8402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545

This theorem is used by:  inf3lemd  9610  inf3lem1  9611  inf3lem2  9612  inf3lem3  9613