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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 3203 | . . 3 ⊢ 𝑦 ∈ V | |
2 | 1 | elima 5471 | . 2 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦) |
3 | df-br 4654 | . . . 4 ⊢ (𝑏∩ 𝐴𝑦 ↔ 〈𝑏, 𝑦〉 ∈ ∩ 𝐴) | |
4 | opex 4932 | . . . . 5 ⊢ 〈𝑏, 𝑦〉 ∈ V | |
5 | 4 | elint2 4482 | . . . 4 ⊢ (〈𝑏, 𝑦〉 ∈ ∩ 𝐴 ↔ ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
6 | 3, 5 | bitri 264 | . . 3 ⊢ (𝑏∩ 𝐴𝑦 ↔ ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
7 | 6 | rexbii 3041 | . 2 ⊢ (∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦 ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
8 | 2, 7 | bitri 264 | 1 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 〈cop 4183 ∩ cint 4475 class class class wbr 4653 “ cima 5117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-int 4476 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 |
This theorem is referenced by: intimass 37946 intimag 37948 |
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