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

Theorem inf3lema 9075
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9086 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 4134 . . 3 (𝑓 = 𝐴 → (𝑓𝑥) = (𝐴𝑥))
21sseq1d 3949 . 2 (𝑓 = 𝐴 → ((𝑓𝑥) ⊆ 𝐵 ↔ (𝐴𝑥) ⊆ 𝐵))
3 inf3lem.4 . . 3 𝐵 ∈ V
4 sseq2 3944 . . . . 5 (𝑣 = 𝐵 → ((𝑓𝑥) ⊆ 𝑣 ↔ (𝑓𝑥) ⊆ 𝐵))
54rabbidv 3430 . . . 4 (𝑣 = 𝐵 → {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵})
6 inf3lem.1 . . . . 5 𝐺 = (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦})
7 sseq2 3944 . . . . . . . 8 (𝑦 = 𝑣 → ((𝑤𝑥) ⊆ 𝑦 ↔ (𝑤𝑥) ⊆ 𝑣))
87rabbidv 3430 . . . . . . 7 (𝑦 = 𝑣 → {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦} = {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑣})
9 ineq1 4134 . . . . . . . . 9 (𝑤 = 𝑓 → (𝑤𝑥) = (𝑓𝑥))
109sseq1d 3949 . . . . . . . 8 (𝑤 = 𝑓 → ((𝑤𝑥) ⊆ 𝑣 ↔ (𝑓𝑥) ⊆ 𝑣))
1110cbvrabv 3442 . . . . . . 7 {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑣} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣}
128, 11eqtrdi 2852 . . . . . 6 (𝑦 = 𝑣 → {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦} = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
1312cbvmptv 5136 . . . . 5 (𝑦 ∈ V ↦ {𝑤𝑥 ∣ (𝑤𝑥) ⊆ 𝑦}) = (𝑣 ∈ V ↦ {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
146, 13eqtri 2824 . . . 4 𝐺 = (𝑣 ∈ V ↦ {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝑣})
15 vex 3447 . . . . 5 𝑥 ∈ V
1615rabex 5202 . . . 4 {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵} ∈ V
175, 14, 16fvmpt 6749 . . 3 (𝐵 ∈ V → (𝐺𝐵) = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵})
183, 17ax-mp 5 . 2 (𝐺𝐵) = {𝑓𝑥 ∣ (𝑓𝑥) ⊆ 𝐵}
192, 18elrab2 3634 1 (𝐴 ∈ (𝐺𝐵) ↔ (𝐴𝑥 ∧ (𝐴𝑥) ⊆ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2112  {crab 3113  Vcvv 3444  cin 3883  wss 3884  c0 4246  cmpt 5113  cres 5525  cfv 6328  ωcom 7564  reccrdg 8032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336
This theorem is referenced by:  inf3lemd  9078  inf3lem1  9079  inf3lem2  9080  inf3lem3  9081
  Copyright terms: Public domain W3C validator