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Theorem elimaint 41567
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 3445 . . 3 𝑦 ∈ V
21elima 5998 . 2 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵 𝑏 𝐴𝑦)
3 df-br 5090 . . . 4 (𝑏 𝐴𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝐴)
4 opex 5403 . . . . 5 𝑏, 𝑦⟩ ∈ V
54elint2 4900 . . . 4 (⟨𝑏, 𝑦⟩ ∈ 𝐴 ↔ ∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
63, 5bitri 274 . . 3 (𝑏 𝐴𝑦 ↔ ∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
76rexbii 3093 . 2 (∃𝑏𝐵 𝑏 𝐴𝑦 ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
82, 7bitri 274 1 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2105  wral 3061  wrex 3070  cop 4578   cint 4893   class class class wbr 5089  cima 5617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-int 4894  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627
This theorem is referenced by:  intimass  41572  intimag  41574
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