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Theorem inf3lema 9567
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9578 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 4170 . . 3 (𝑓 = 𝐴 → (𝑓𝑥) = (𝐴𝑥))
21sseq1d 3980 . 2 (𝑓 = 𝐴 → ((𝑓𝑥) ⊆ 𝐵 ↔ (𝐴𝑥) ⊆ 𝐵))
3 inf3lem.4 . . 3 𝐵 ∈ V
4 sseq2 3975 . . . . 5 (𝑣 = 𝐵 → ((𝑓𝑥) ⊆ 𝑣 ↔ (𝑓𝑥) ⊆ 𝐵))
54rabbidv 3418 . . . 4 (𝑣 = 𝐵 → {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵})
6 inf3lem.1 . . . . 5 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
7 sseq2 3975 . . . . . . . 8 (𝑦 = 𝑣 → ((𝑤𝑥) ⊆ 𝑦 ↔ (𝑤𝑥) ⊆ 𝑣))
87rabbidv 3418 . . . . . . 7 (𝑦 = 𝑣 → {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦} = {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑣})
9 ineq1 4170 . . . . . . . . 9 (𝑤 = 𝑓 → (𝑤𝑥) = (𝑓𝑥))
109sseq1d 3980 . . . . . . . 8 (𝑤 = 𝑓 → ((𝑤𝑥) ⊆ 𝑣 ↔ (𝑓𝑥) ⊆ 𝑣))
1110cbvrabv 3420 . . . . . . 7 {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑣} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣}
128, 11eqtrdi 2793 . . . . . 6 (𝑦 = 𝑣 → {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
1312cbvmptv 5223 . . . . 5 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑣 ∈ V ↦ {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
146, 13eqtri 2765 . . . 4 𝐺 = (𝑣 ∈ V ↦ {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
15 vex 3452 . . . . 5 𝑥 ∈ V
1615rabex 5294 . . . 4 {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵} ∈ V
175, 14, 16fvmpt 6953 . . 3 (𝐵 ∈ V → (𝐺𝐵) = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵})
183, 17ax-mp 5 . 2 (𝐺𝐵) = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵}
192, 18elrab2 3653 1 (𝐴 ∈ (𝐺𝐵) ↔ (𝐴𝑥 ∧ (𝐴𝑥) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3410  Vcvv 3448  cin 3914  wss 3915  c0 4287  cmpt 5193  cres 5640  cfv 6501  ωcom 7807  reccrdg 8360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509
This theorem is referenced by:  inf3lemd  9570  inf3lem1  9571  inf3lem2  9572  inf3lem3  9573
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