Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elimaint Structured version   Visualization version   GIF version

Theorem elimaint 39978
Description: Element of image of intersection. (Contributed by RP, 13-Apr-2020.)
Assertion
Ref Expression
elimaint (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
Distinct variable groups:   𝑎,𝑏,𝐴   𝐵,𝑏   𝑦,𝑎,𝑏
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑦,𝑎)

Proof of Theorem elimaint
StepHypRef Expression
1 vex 3496 . . 3 𝑦 ∈ V
21elima 5927 . 2 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵 𝑏 𝐴𝑦)
3 df-br 5058 . . . 4 (𝑏 𝐴𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝐴)
4 opex 5347 . . . . 5 𝑏, 𝑦⟩ ∈ V
54elint2 4874 . . . 4 (⟨𝑏, 𝑦⟩ ∈ 𝐴 ↔ ∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
63, 5bitri 277 . . 3 (𝑏 𝐴𝑦 ↔ ∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
76rexbii 3245 . 2 (∃𝑏𝐵 𝑏 𝐴𝑦 ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
82, 7bitri 277 1 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wcel 2107  wral 3136  wrex 3137  cop 4565   cint 4867   class class class wbr 5057  cima 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-int 4868  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by:  intimass  39984  intimag  39986
  Copyright terms: Public domain W3C validator