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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 6042 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 6039 | . 2 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵) | |
2 | disjsn 4647 | . . 3 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴) | |
3 | eqid 2821 | . . 3 ⊢ 𝐵 = 𝐵 | |
4 | 2, 3 | ifbieq2i 4491 | . 2 ⊢ if((𝐴 ∩ {𝑋}) = ∅, ∅, 𝐵) = if(¬ 𝑋 ∈ 𝐴, ∅, 𝐵) |
5 | ifnot 4517 | . 2 ⊢ if(¬ 𝑋 ∈ 𝐴, ∅, 𝐵) = if(𝑋 ∈ 𝐴, 𝐵, ∅) | |
6 | 1, 4, 5 | 3eqtri 2848 | 1 ⊢ ((𝐴 × 𝐵) “ {𝑋}) = if(𝑋 ∈ 𝐴, 𝐵, ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ∅c0 4291 ifcif 4467 {csn 4567 × cxp 5553 “ cima 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: bj-xpima1sn 34271 bj-xpima2sn 34273 |
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