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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-xpimasn | Structured version Visualization version GIF version |
Description: The image of a singleton, general case. [Change and relabel xpimasn 6191 accordingly, maybe to xpima2sn.] (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-xpimasn | ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpima 6188 | . 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵) | |
2 | disjsn 4717 | . . 3 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴) | |
3 | eqid 2725 | . . 3 ⊢ 𝐵 = 𝐵 | |
4 | 2, 3 | ifbieq2i 4555 | . 2 ⊢ if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵) = if(¬ 𝑋 ∈ 𝐴, ∅, 𝐵) |
5 | ifnot 4582 | . 2 ⊢ if(¬ 𝑋 ∈ 𝐴, ∅, 𝐵) = if(𝑋 ∈ 𝐴, 𝐵, ∅) | |
6 | 1, 4, 5 | 3eqtri 2757 | 1 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 ∩ cin 3943 ∅c0 4322 ifcif 4530 {csn 4630 × cxp 5676 “ cima 5681 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 |
This theorem is referenced by: bj-xpima1sn 36586 bj-xpima2sn 36588 |
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