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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 3418 | . . 3 ⊢ 𝑦 ∈ V | |
2 | 1 | elima 5713 | . 2 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦) |
3 | df-br 4875 | . . . 4 ⊢ (𝑏∩ 𝐴𝑦 ↔ 〈𝑏, 𝑦〉 ∈ ∩ 𝐴) | |
4 | opex 5154 | . . . . 5 ⊢ 〈𝑏, 𝑦〉 ∈ V | |
5 | 4 | elint2 4705 | . . . 4 ⊢ (〈𝑏, 𝑦〉 ∈ ∩ 𝐴 ↔ ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
6 | 3, 5 | bitri 267 | . . 3 ⊢ (𝑏∩ 𝐴𝑦 ↔ ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
7 | 6 | rexbii 3252 | . 2 ⊢ (∃𝑏 ∈ 𝐵 𝑏∩ 𝐴𝑦 ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
8 | 2, 7 | bitri 267 | 1 ⊢ (𝑦 ∈ (∩ 𝐴 “ 𝐵) ↔ ∃𝑏 ∈ 𝐵 ∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2166 ∀wral 3118 ∃wrex 3119 〈cop 4404 ∩ cint 4698 class class class wbr 4874 “ cima 5346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-int 4699 df-br 4875 df-opab 4937 df-xp 5349 df-cnv 5351 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 |
This theorem is referenced by: intimass 38788 intimag 38790 |
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