Theorem elimaint 38782

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 3418	. . 3 ⊢ 𝑦 ∈ V
2	1	elima 5713	. 2 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦)
3		df-br 4875	. . . 4 ⊢ (𝑏∩ 𝐴𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ ∩ 𝐴)
4		opex 5154	. . . . 5 ⊢ ⟨𝑏, 𝑦⟩ ∈ V
5	4	elint2 4705	. . . 4 ⊢ (⟨𝑏, 𝑦⟩ ∈ ∩ 𝐴 ↔ ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
6	3, 5	bitri 267	. . 3 ⊢ (𝑏∩ 𝐴𝑦 ↔ ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
7	6	rexbii 3252	. 2 ⊢ (∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦 ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)
8	2, 7	bitri 267	1 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎)

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