Theorem inf3lema 9628

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9639 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 4110	. . 3 ⊢ (𝑓 = 𝐴 → (𝑓 ∩ 𝑥) = (𝐴 ∩ 𝑥))
2	1	sseq1d 3939	. 2 ⊢ (𝑓 = 𝐴 → ((𝑓 ∩ 𝑥) ⊆ 𝐵 ↔ (𝐴 ∩ 𝑥) ⊆ 𝐵))
3		inf3lem.4	. . 3 ⊢ 𝐵 ∈ V
4		sseq2 3934	. . . . 5 ⊢ (𝑣 = 𝐵 → ((𝑓 ∩ 𝑥) ⊆ 𝑣 ↔ (𝑓 ∩ 𝑥) ⊆ 𝐵))
5	4	rabbidv 3419	. . . 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 3934	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑤 ∩ 𝑥) ⊆ 𝑦 ↔ (𝑤 ∩ 𝑥) ⊆ 𝑣))
8	7	rabbidv 3419	. . . . . . 7 ⊢ (𝑦 = 𝑣 → {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦} = {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑣})
9		ineq1 4110	. . . . . . . . 9 ⊢ (𝑤 = 𝑓 → (𝑤 ∩ 𝑥) = (𝑓 ∩ 𝑥))
10	9	sseq1d 3939	. . . . . . . 8 ⊢ (𝑤 = 𝑓 → ((𝑤 ∩ 𝑥) ⊆ 𝑣 ↔ (𝑓 ∩ 𝑥) ⊆ 𝑣))
11	10	cbvrabv 3422	. . . . . . 7 ⊢ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑣} = {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣}
12	8, 11	eqtrdi 2811	. . . . . 6 ⊢ (𝑦 = 𝑣 → {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦} = {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
13	12	cbvmptv 5208	. . . . 5 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑣 ∈ V ↦ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
14	6, 13	eqtri 2783	. . . 4 ⊢ 𝐺 = (𝑣 ∈ V ↦ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
15		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
16	15	rabex 5299	. . . 4 ⊢ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝐵} ∈ V
17	5, 14, 16	fvmpt 6993	. . 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 3648	1 ⊢ (𝐴 ∈ (𝐺‘𝐵) ↔ (𝐴 ∈ 𝑥 ∧ (𝐴 ∩ 𝑥) ⊆ 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278   ↦ cmpt 5185   ↾ cres 5652  ‘cfv 6538  ωcom 7877  reccrdg 8420

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is used by:  inf3lemd  9631  inf3lem1  9632  inf3lem2  9633  inf3lem3  9634