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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 1839 | . 2 ⊢ (𝐵 ∈ 𝑉 → ∀𝑦 𝐵 ∈ 𝑉) | |
2 | r19.12sn 4255 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → (∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎 ↔ ∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}〈𝑏, 𝑦〉 ∈ 𝑎)) | |
3 | 2 | biimprd 238 | . . 3 ⊢ (𝐵 ∈ 𝑉 → (∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎)) |
4 | 3 | alimi 1739 | . 2 ⊢ (∀𝑦 𝐵 ∈ 𝑉 → ∀𝑦(∀𝑎 ∈ 𝐴 ∃𝑏 ∈ {𝐵}〈𝑏, 𝑦〉 ∈ 𝑎 → ∃𝑏 ∈ {𝐵}∀𝑎 ∈ 𝐴 〈𝑏, 𝑦〉 ∈ 𝑎)) |
5 | intimag 37948 | . 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 1481 = wceq 1483 ∈ wcel 1990 {cab 2608 ∀wral 2912 ∃wrex 2913 {csn 4177 〈cop 4183 ∩ cint 4475 “ cima 5117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 |
This theorem is referenced by: intimasn2 37950 brtrclfv2 38019 |
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