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Theorem elimaint 38782
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 3418 . . 3 𝑦 ∈ V
21elima 5713 . 2 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵 𝑏 𝐴𝑦)
3 df-br 4875 . . . 4 (𝑏 𝐴𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ 𝐴)
4 opex 5154 . . . . 5 𝑏, 𝑦⟩ ∈ V
54elint2 4705 . . . 4 (⟨𝑏, 𝑦⟩ ∈ 𝐴 ↔ ∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
63, 5bitri 267 . . 3 (𝑏 𝐴𝑦 ↔ ∀𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
76rexbii 3252 . 2 (∃𝑏𝐵 𝑏 𝐴𝑦 ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
82, 7bitri 267 1 (𝑦 ∈ ( 𝐴𝐵) ↔ ∃𝑏𝐵𝑎𝐴𝑏, 𝑦⟩ ∈ 𝑎)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wcel 2166  wral 3118  wrex 3119  cop 4404   cint 4698   class class class wbr 4874  cima 5346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-int 4699  df-br 4875  df-opab 4937  df-xp 5349  df-cnv 5351  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356
This theorem is referenced by:  intimass  38788  intimag  38790
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