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Mirrors > Home > ILE Home > Th. List > xpimasn | GIF version |
Description: The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
Ref | Expression |
---|---|
xpimasn | ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snmg 3636 | . . 3 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 𝑥 ∈ {𝑋}) | |
2 | snssi 3659 | . . . . . 6 ⊢ (𝑋 ∈ 𝐴 → {𝑋} ⊆ 𝐴) | |
3 | dfss1 3275 | . . . . . 6 ⊢ ({𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑋}) = {𝑋}) | |
4 | 2, 3 | sylib 121 | . . . . 5 ⊢ (𝑋 ∈ 𝐴 → (𝐴 ∩ {𝑋}) = {𝑋}) |
5 | 4 | eleq2d 2207 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ 𝑥 ∈ {𝑋})) |
6 | 5 | exbidv 1797 | . . 3 ⊢ (𝑋 ∈ 𝐴 → (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ ∃𝑥 𝑥 ∈ {𝑋})) |
7 | 1, 6 | mpbird 166 | . 2 ⊢ (𝑋 ∈ 𝐴 → ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋})) |
8 | xpima2m 4981 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}) → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) | |
9 | 7, 8 | syl 14 | 1 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∩ cin 3065 ⊆ wss 3066 {csn 3522 × cxp 4532 “ cima 4537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-xp 4540 df-rel 4541 df-cnv 4542 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 |
This theorem is referenced by: imasnopn 12457 |
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