Theorem inf3lema 9594

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9605 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 4167	. . 3 ⊢ (𝑓 = 𝐴 → (𝑓 ∩ 𝑥) = (𝐴 ∩ 𝑥))
2	1	sseq1d 3969	. 2 ⊢ (𝑓 = 𝐴 → ((𝑓 ∩ 𝑥) ⊆ 𝐵 ↔ (𝐴 ∩ 𝑥) ⊆ 𝐵))
3		inf3lem.4	. . 3 ⊢ 𝐵 ∈ V
4		sseq2 3964	. . . . 5 ⊢ (𝑣 = 𝐵 → ((𝑓 ∩ 𝑥) ⊆ 𝑣 ↔ (𝑓 ∩ 𝑥) ⊆ 𝐵))
5	4	rabbidv 3423	. . . 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 3964	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑤 ∩ 𝑥) ⊆ 𝑦 ↔ (𝑤 ∩ 𝑥) ⊆ 𝑣))
8	7	rabbidv 3423	. . . . . . 7 ⊢ (𝑦 = 𝑣 → {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦} = {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑣})
9		ineq1 4167	. . . . . . . . 9 ⊢ (𝑤 = 𝑓 → (𝑤 ∩ 𝑥) = (𝑓 ∩ 𝑥))
10	9	sseq1d 3969	. . . . . . . 8 ⊢ (𝑤 = 𝑓 → ((𝑤 ∩ 𝑥) ⊆ 𝑣 ↔ (𝑓 ∩ 𝑥) ⊆ 𝑣))
11	10	cbvrabv 3426	. . . . . . 7 ⊢ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑣} = {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣}
12	8, 11	eqtrdi 2814	. . . . . 6 ⊢ (𝑦 = 𝑣 → {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦} = {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
13	12	cbvmptv 5216	. . . . 5 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑣 ∈ V ↦ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
14	6, 13	eqtri 2786	. . . 4 ⊢ 𝐺 = (𝑣 ∈ V ↦ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
15		vex 3459	. . . . 5 ⊢ 𝑥 ∈ V
16	15	rabex 5311	. . . 4 ⊢ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝐵} ∈ V
17	5, 14, 16	fvmpt 6991	. . 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 3655	1 ⊢ (𝐴 ∈ (𝐺‘𝐵) ↔ (𝐴 ∈ 𝑥 ∧ (𝐴 ∩ 𝑥) ⊆ 𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   ∩ cin 3905   ⊆ wss 3906  ∅c0 4287   ↦ cmpt 5193   ↾ cres 5665  ‘cfv 6538  ωcom 7863  reccrdg 8397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is referenced by:  inf3lemd  9597  inf3lem1  9598  inf3lem2  9599  inf3lem3  9600