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| Mirrors > Home > MPE Home > Th. List > xpimasn | Structured version Visualization version GIF version | ||
| Description: Direct 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 4711 | . . . 4 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴) | |
| 2 | 1 | necon3abii 2987 | . . 3 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ ¬ ¬ 𝑋 ∈ 𝐴) |
| 3 | notnotb 315 | . . 3 ⊢ (𝑋 ∈ 𝐴 ↔ ¬ ¬ 𝑋 ∈ 𝐴) | |
| 4 | 2, 3 | bitr4i 278 | . 2 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ 𝑋 ∈ 𝐴) |
| 5 | xpima2 6204 | . 2 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) | |
| 6 | 4, 5 | sylbir 235 | 1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∩ cin 3950 ∅c0 4333 {csn 4626 × cxp 5683 “ cima 5688 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: imasnopn 23698 imasncld 23699 imasncls 23700 restutopopn 24247 arearect 43227 |
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