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Theorem inf3lema 8404
 Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. (Contributed by NM, 28-Oct-1996.)
Hypotheses
Ref Expression
inf3lem.1 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
inf3lem.2 𝐹 = (rec(𝐺, ∅) ↾ ω)
inf3lem.3 𝐴 ∈ V
inf3lem.4 𝐵 ∈ V
Assertion
Ref Expression
inf3lema (𝐴 ∈ (𝐺𝐵) ↔ (𝐴𝑥 ∧ (𝐴𝑥) ⊆ 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.
StepHypRef Expression
1 ineq1 3769 . . 3 (𝑓 = 𝐴 → (𝑓𝑥) = (𝐴𝑥))
21sseq1d 3595 . 2 (𝑓 = 𝐴 → ((𝑓𝑥) ⊆ 𝐵 ↔ (𝐴𝑥) ⊆ 𝐵))
3 inf3lem.4 . . 3 𝐵 ∈ V
4 sseq2 3590 . . . . 5 (𝑣 = 𝐵 → ((𝑓𝑥) ⊆ 𝑣 ↔ (𝑓𝑥) ⊆ 𝐵))
54rabbidv 3164 . . . 4 (𝑣 = 𝐵 → {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵})
6 inf3lem.1 . . . . 5 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
7 sseq2 3590 . . . . . . . 8 (𝑦 = 𝑣 → ((𝑤𝑥) ⊆ 𝑦 ↔ (𝑤𝑥) ⊆ 𝑣))
87rabbidv 3164 . . . . . . 7 (𝑦 = 𝑣 → {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦} = {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑣})
9 ineq1 3769 . . . . . . . . 9 (𝑤 = 𝑓 → (𝑤𝑥) = (𝑓𝑥))
109sseq1d 3595 . . . . . . . 8 (𝑤 = 𝑓 → ((𝑤𝑥) ⊆ 𝑣 ↔ (𝑓𝑥) ⊆ 𝑣))
1110cbvrabv 3172 . . . . . . 7 {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑣} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣}
128, 11syl6eq 2660 . . . . . 6 (𝑦 = 𝑣 → {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
1312cbvmptv 4678 . . . . 5 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑣 ∈ V ↦ {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
146, 13eqtri 2632 . . . 4 𝐺 = (𝑣 ∈ V ↦ {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
15 vex 3176 . . . . 5 𝑥 ∈ V
1615rabex 4740 . . . 4 {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵} ∈ V
175, 14, 16fvmpt 6191 . . 3 (𝐵 ∈ V → (𝐺𝐵) = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵})
183, 17ax-mp 5 . 2 (𝐺𝐵) = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵}
192, 18elrab2 3333 1 (𝐴 ∈ (𝐺𝐵) ↔ (𝐴𝑥 ∧ (𝐴𝑥) ⊆ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   ↦ cmpt 4643   ↾ cres 5040  ‘cfv 5804  ωcom 6957  reccrdg 7392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  inf3lemd  8407  inf3lem1  8408  inf3lem2  8409  inf3lem3  8410
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