Theorem elimaint 44267

Description: Element of image of intersection. (Contributed by RP, 13-Apr-2020.)

Assertion

Ref	Expression
elimaint	⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 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

Step	Hyp	Ref	Expression
1		vex 3467	. . 3 ⊢ 𝑦 ∈ V
2	1	elima 6068	. 2 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦)
3		df-br 5114	. . . 4 ⊢ (𝑏∩ 𝐴𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ ∩ 𝐴)
4		opex 5446	. . . . 5 ⊢ ⟨𝑏, 𝑦⟩ ∈ V
5	4	elint2 4923	. . . 4 ⊢ (⟨𝑏, 𝑦⟩ ∈ ∩ 𝐴 ↔ ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
6	3, 5	bitri 278	. . 3 ⊢ (𝑏∩ 𝐴𝑦 ↔ ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
7	6	rexbii 3118	. 2 ⊢ (∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦 ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
8	2, 7	bitri 278	1 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)

Syntax hints:   ↔ wb 209   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  ⟨cop 4600  ∩ cint 4916   class class class wbr 5113   “ cima 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-int 4917  df-br 5114  df-opab 5178  df-xp 5668  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675

This theorem is referenced by:  intimass  44272  intimag  44274