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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 3497 | . . 3 ⊢ 𝑦 ∈ V | |
2 | 1 | elima 5928 | . 2 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦) |
3 | df-br 5059 | . . . 4 ⊢ (𝑏∩ 𝐴𝑦 ↔ 〈𝑏, 𝑦〉 ∈ ∩ 𝐴) | |
4 | opex 5348 | . . . . 5 ⊢ 〈𝑏, 𝑦〉 ∈ V | |
5 | 4 | elint2 4875 | . . . 4 ⊢ (〈𝑏, 𝑦〉 ∈ ∩ 𝐴 ↔ ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
6 | 3, 5 | bitri 277 | . . 3 ⊢ (𝑏∩ 𝐴𝑦 ↔ ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
7 | 6 | rexbii 3247 | . 2 ⊢ (∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦 ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
8 | 2, 7 | bitri 277 | 1 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 〈cop 4566 ∩ cint 4868 class class class wbr 5058 “ cima 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-int 4869 df-br 5059 df-opab 5121 df-xp 5555 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 |
This theorem is referenced by: intimass 39992 intimag 39994 |
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