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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 3467 | . . 3 ⊢ 𝑦 ∈ V | |
2 | 1 | elima 6065 | . 2 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦) |
3 | df-br 5145 | . . . 4 ⊢ (𝑏∩ 𝐴𝑦 ↔ 〈𝑏, 𝑦〉 ∈ ∩ 𝐴) | |
4 | opex 5461 | . . . . 5 ⊢ 〈𝑏, 𝑦〉 ∈ V | |
5 | 4 | elint2 4954 | . . . 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 3084 | . 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 2099 ∀wral 3051 ∃wrex 3060 〈cop 4630 ∩ cint 4947 class class class wbr 5144 “ cima 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-int 4948 df-br 5145 df-opab 5207 df-xp 5679 df-cnv 5681 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 |
This theorem is referenced by: intimass 43356 intimag 43358 |
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