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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 5835 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 5832 | . 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵) | |
2 | disjsn 4478 | . . 3 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴) | |
3 | eqid 2778 | . . 3 ⊢ 𝐵 = 𝐵 | |
4 | 2, 3 | ifbieq2i 4331 | . 2 ⊢ if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵) = if(¬ 𝑋 ∈ 𝐴, ∅, 𝐵) |
5 | ifnot 4357 | . 2 ⊢ if(¬ 𝑋 ∈ 𝐴, ∅, 𝐵) = if(𝑋 ∈ 𝐴, 𝐵, ∅) | |
6 | 1, 4, 5 | 3eqtri 2806 | 1 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1601 ∈ wcel 2107 ∩ cin 3791 ∅c0 4141 ifcif 4307 {csn 4398 × cxp 5355 “ cima 5360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-br 4889 df-opab 4951 df-xp 5363 df-rel 5364 df-cnv 5365 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 |
This theorem is referenced by: bj-xpima1sn 33519 bj-xpima2sn 33521 |
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