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Mirrors > Home > MPE Home > Th. List > xpimasn | Structured version Visualization version GIF version |
Description: The image of a singleton by a Cartesian product. (Contributed by Thierry Arnoux, 14-Jan-2018.) (Proof shortened by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
xpimasn | ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjsn 4192 | . . . 4 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴) | |
2 | 1 | necon3abii 2828 | . . 3 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ ¬ ¬ 𝑋 ∈ 𝐴) |
3 | notnotb 303 | . . 3 ⊢ (𝑋 ∈ 𝐴 ↔ ¬ ¬ 𝑋 ∈ 𝐴) | |
4 | 2, 3 | bitr4i 266 | . 2 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ 𝑋 ∈ 𝐴) |
5 | xpima2 5497 | . 2 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) | |
6 | 4, 5 | sylbir 224 | 1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∩ cin 3539 ∅c0 3874 {csn 4125 × cxp 5036 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: imasnopn 21303 imasncld 21304 imasncls 21305 restutopopn 21852 arearect 36820 |
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