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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 4720 | . . . 4 ⊢ ((𝐴 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐴) | |
2 | 1 | necon3abii 2984 | . . 3 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ ¬ ¬ 𝑋 ∈ 𝐴) |
3 | notnotb 314 | . . 3 ⊢ (𝑋 ∈ 𝐴 ↔ ¬ ¬ 𝑋 ∈ 𝐴) | |
4 | 2, 3 | bitr4i 277 | . 2 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ ↔ 𝑋 ∈ 𝐴) |
5 | xpima2 6193 | . 2 ⊢ ((𝐴 ∩ {𝑋}) ≠ ∅ → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) | |
6 | 4, 5 | sylbir 234 | 1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∩ cin 3948 ∅c0 4326 {csn 4632 × cxp 5680 “ cima 5685 |
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 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-xp 5688 df-rel 5689 df-cnv 5690 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 |
This theorem is referenced by: imasnopn 23614 imasncld 23615 imasncls 23616 restutopopn 24163 arearect 42674 |
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