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Theorem elimaint 37460
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 3193 . . 3 𝑦 ∈ V
21elima 5440 . 2 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵 𝑏 𝐴𝑦)
3 df-br 4624 . . . 4 (𝑏 𝐴𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝐴)
4 opex 4903 . . . . 5 𝑏, 𝑦⟩ ∈ V
54elint2 4454 . . . 4 (⟨𝑏, 𝑦⟩ ∈ 𝐴 ↔ ∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
63, 5bitri 264 . . 3 (𝑏 𝐴𝑦 ↔ ∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
76rexbii 3036 . 2 (∃𝑏𝐵 𝑏 𝐴𝑦 ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
82, 7bitri 264 1 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wcel 1987  wral 2908  wrex 2909  cop 4161   cint 4447   class class class wbr 4623  cima 5087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-int 4448  df-br 4624  df-opab 4684  df-xp 5090  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097
This theorem is referenced by:  intimass  37466  intimag  37468
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