Theorem inf3lema 9576

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9587 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 4165	. . 3 ⊢ (𝑓 = 𝐴 → (𝑓 ∩ 𝑥) = (𝐴 ∩ 𝑥))
2	1	sseq1d 3967	. 2 ⊢ (𝑓 = 𝐴 → ((𝑓 ∩ 𝑥) ⊆ 𝐵 ↔ (𝐴 ∩ 𝑥) ⊆ 𝐵))
3		inf3lem.4	. . 3 ⊢ 𝐵 ∈ V
4		sseq2 3962	. . . . 5 ⊢ (𝑣 = 𝐵 → ((𝑓 ∩ 𝑥) ⊆ 𝑣 ↔ (𝑓 ∩ 𝑥) ⊆ 𝐵))
5	4	rabbidv 3420	. . . 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 3962	. . . . . . . 8 ⊢ (𝑦 = 𝑣 → ((𝑤 ∩ 𝑥) ⊆ 𝑦 ↔ (𝑤 ∩ 𝑥) ⊆ 𝑣))
8	7	rabbidv 3420	. . . . . . 7 ⊢ (𝑦 = 𝑣 → {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦} = {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑣})
9		ineq1 4165	. . . . . . . . 9 ⊢ (𝑤 = 𝑓 → (𝑤 ∩ 𝑥) = (𝑓 ∩ 𝑥))
10	9	sseq1d 3967	. . . . . . . 8 ⊢ (𝑤 = 𝑓 → ((𝑤 ∩ 𝑥) ⊆ 𝑣 ↔ (𝑓 ∩ 𝑥) ⊆ 𝑣))
11	10	cbvrabv 3423	. . . . . . 7 ⊢ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑣} = {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣}
12	8, 11	eqtrdi 2812	. . . . . 6 ⊢ (𝑦 = 𝑣 → {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦} = {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
13	12	cbvmptv 5203	. . . . 5 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑣 ∈ V ↦ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
14	6, 13	eqtri 2784	. . . 4 ⊢ 𝐺 = (𝑣 ∈ V ↦ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝑣})
15		vex 3457	. . . . 5 ⊢ 𝑥 ∈ V
16	15	rabex 5294	. . . 4 ⊢ {𝑓 ∈ 𝑥 ∣ (𝑓 ∩ 𝑥) ⊆ 𝐵} ∈ V
17	5, 14, 16	fvmpt 6971	. . 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 3653	1 ⊢ (𝐴 ∈ (𝐺‘𝐵) ↔ (𝐴 ∈ 𝑥 ∧ (𝐴 ∩ 𝑥) ⊆ 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  {crab 3413  Vcvv 3453   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285   ↦ cmpt 5180   ↾ cres 5647  ‘cfv 6517  ωcom 7842  reccrdg 8375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525

This theorem is referenced by:  inf3lemd  9579  inf3lem1  9580  inf3lem2  9581  inf3lem3  9582