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Theorem elimaint 42973
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 3472 . . 3 𝑦 ∈ V
21elima 6058 . 2 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵 𝑏 𝐴𝑦)
3 df-br 5142 . . . 4 (𝑏 𝐴𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝐴)
4 opex 5457 . . . . 5 𝑏, 𝑦⟩ ∈ V
54elint2 4950 . . . 4 (⟨𝑏, 𝑦⟩ ∈ 𝐴 ↔ ∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
63, 5bitri 275 . . 3 (𝑏 𝐴𝑦 ↔ ∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
76rexbii 3088 . 2 (∃𝑏𝐵 𝑏 𝐴𝑦 ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
82, 7bitri 275 1 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098  wral 3055  wrex 3064  cop 4629   cint 4943   class class class wbr 5141  cima 5672
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-int 4944  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682
This theorem is referenced by:  intimass  42978  intimag  42980
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