MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  inf3lema Structured version   Visualization version   GIF version

Theorem inf3lema 9619
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9630 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 4206 . . 3 (𝑓 = 𝐴 → (𝑓𝑥) = (𝐴𝑥))
21sseq1d 4014 . 2 (𝑓 = 𝐴 → ((𝑓𝑥) ⊆ 𝐵 ↔ (𝐴𝑥) ⊆ 𝐵))
3 inf3lem.4 . . 3 𝐵 ∈ V
4 sseq2 4009 . . . . 5 (𝑣 = 𝐵 → ((𝑓𝑥) ⊆ 𝑣 ↔ (𝑓𝑥) ⊆ 𝐵))
54rabbidv 3441 . . . 4 (𝑣 = 𝐵 → {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵})
6 inf3lem.1 . . . . 5 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
7 sseq2 4009 . . . . . . . 8 (𝑦 = 𝑣 → ((𝑤𝑥) ⊆ 𝑦 ↔ (𝑤𝑥) ⊆ 𝑣))
87rabbidv 3441 . . . . . . 7 (𝑦 = 𝑣 → {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦} = {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑣})
9 ineq1 4206 . . . . . . . . 9 (𝑤 = 𝑓 → (𝑤𝑥) = (𝑓𝑥))
109sseq1d 4014 . . . . . . . 8 (𝑤 = 𝑓 → ((𝑤𝑥) ⊆ 𝑣 ↔ (𝑓𝑥) ⊆ 𝑣))
1110cbvrabv 3443 . . . . . . 7 {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑣} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣}
128, 11eqtrdi 2789 . . . . . 6 (𝑦 = 𝑣 → {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
1312cbvmptv 5262 . . . . 5 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑣 ∈ V ↦ {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
146, 13eqtri 2761 . . . 4 𝐺 = (𝑣 ∈ V ↦ {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
15 vex 3479 . . . . 5 𝑥 ∈ V
1615rabex 5333 . . . 4 {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵} ∈ V
175, 14, 16fvmpt 6999 . . 3 (𝐵 ∈ V → (𝐺𝐵) = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵})
183, 17ax-mp 5 . 2 (𝐺𝐵) = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵}
192, 18elrab2 3687 1 (𝐴 ∈ (𝐺𝐵) ↔ (𝐴𝑥 ∧ (𝐴𝑥) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3433  Vcvv 3475  cin 3948  wss 3949  c0 4323  cmpt 5232  cres 5679  cfv 6544  ωcom 7855  reccrdg 8409
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552
This theorem is referenced by:  inf3lemd  9622  inf3lem1  9623  inf3lem2  9624  inf3lem3  9625
  Copyright terms: Public domain W3C validator