Theorem elimaint 43613

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 3492	. . 3 ⊢ 𝑦 ∈ V
2	1	elima 6096	. 2 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦)
3		df-br 5167	. . . 4 ⊢ (𝑏∩ 𝐴𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ ∩ 𝐴)
4		opex 5484	. . . . 5 ⊢ ⟨𝑏, 𝑦⟩ ∈ V
5	4	elint2 4977	. . . 4 ⊢ (⟨𝑏, 𝑦⟩ ∈ ∩ 𝐴 ↔ ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
6	3, 5	bitri 275	. . 3 ⊢ (𝑏∩ 𝐴𝑦 ↔ ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
7	6	rexbii 3100	. 2 ⊢ (∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦 ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
8	2, 7	bitri 275	1 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  ⟨cop 4654  ∩ cint 4970   class class class wbr 5166   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-int 4971  df-br 5167  df-opab 5229  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  intimass  43618  intimag  43620