Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > elimaint | Structured version Visualization version GIF version |
Description: Element of image of intersection. (Contributed by RP, 13-Apr-2020.) |
Ref | Expression |
---|---|
elimaint | ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3445 | . . 3 ⊢ 𝑦 ∈ V | |
2 | 1 | elima 5998 | . 2 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦) |
3 | df-br 5090 | . . . 4 ⊢ (𝑏∩ 𝐴𝑦 ↔ ⟨𝑏, 𝑦⟩ ∈ ∩ 𝐴) | |
4 | opex 5403 | . . . . 5 ⊢ ⟨𝑏, 𝑦⟩ ∈ V | |
5 | 4 | elint2 4900 | . . . 4 ⊢ (⟨𝑏, 𝑦⟩ ∈ ∩ 𝐴 ↔ ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) |
6 | 3, 5 | bitri 274 | . . 3 ⊢ (𝑏∩ 𝐴𝑦 ↔ ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) |
7 | 6 | rexbii 3093 | . 2 ⊢ (∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦 ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) |
8 | 2, 7 | bitri 274 | 1 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 ⟨𝑏, 𝑦⟩ ∈ 𝑎) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2105 ∀wral 3061 ∃wrex 3070 ⟨cop 4578 ∩ cint 4893 class class class wbr 5089 “ cima 5617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 ax-sep 5240 ax-nul 5247 ax-pr 5369 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4269 df-if 4473 df-sn 4573 df-pr 4575 df-op 4579 df-int 4894 df-br 5090 df-opab 5152 df-xp 5620 df-cnv 5622 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 |
This theorem is referenced by: intimass 41572 intimag 41574 |
Copyright terms: Public domain | W3C validator |