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Mirrors > Home > MPE Home > Th. List > Mathboxes > intimasn | Structured version Visualization version GIF version |
Description: Two ways to express the image of a singleton when the relation is an intersection. (Contributed by RP, 13-Apr-2020.) |
Ref | Expression |
---|---|
intimasn | ⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-5 1907 | . 2 ⊢ (𝐵 ∈ 𝑉 → ∀𝑦 𝐵 ∈ 𝑉) | |
2 | r19.12sn 4649 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎 ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}〈𝑏, 𝑦〉 ∈ 𝑎)) | |
3 | 2 | biimprd 250 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎)) |
4 | 3 | alimi 1808 | . 2 ⊢ (∀𝑦 𝐵 ∈ 𝑉 → ∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎)) |
5 | intimag 39994 | . 2 ⊢ (∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎) → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})}) | |
6 | 1, 4, 5 | 3syl 18 | 1 ⊢ (𝐵 ∈ 𝑉 → (∩ 𝐴 “ {𝐵}) = ∩ {𝑥 ∣ ∃𝑎 ∈ 𝐴 𝑥 = (𝑎 “ {𝐵})}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 = wceq 1533 ∈ wcel 2110 {cab 2799 ∀wral 3138 ∃wrex 3139 {csn 4560 〈cop 4566 ∩ cint 4868 “ 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 ax-un 7455 |
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-sbc 3772 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-uni 4832 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: intimasn2 39996 brtrclfv2 40065 |
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