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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 5934 | . 2 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦) |
3 | df-br 5067 | . . . 4 ⊢ (𝑏∩ 𝐴𝑦 ↔ 〈𝑏, 𝑦〉 ∈ ∩ 𝐴) | |
4 | opex 5356 | . . . . 5 ⊢ 〈𝑏, 𝑦〉 ∈ V | |
5 | 4 | elint2 4883 | . . . 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 2114 ∀wral 3138 ∃wrex 3139 〈cop 4573 ∩ cint 4876 class class class wbr 5066 “ cima 5558 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-int 4877 df-br 5067 df-opab 5129 df-xp 5561 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: intimass 40019 intimag 40021 |
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