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Theorem xpimasn 4894
Description: The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Assertion
Ref Expression
xpimasn (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)

Proof of Theorem xpimasn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 snmg 3566 . . 3 (𝑋𝐴 → ∃𝑥 𝑥 ∈ {𝑋})
2 snssi 3589 . . . . . 6 (𝑋𝐴 → {𝑋} ⊆ 𝐴)
3 dfss1 3207 . . . . . 6 ({𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑋}) = {𝑋})
42, 3sylib 121 . . . . 5 (𝑋𝐴 → (𝐴 ∩ {𝑋}) = {𝑋})
54eleq2d 2158 . . . 4 (𝑋𝐴 → (𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ 𝑥 ∈ {𝑋}))
65exbidv 1754 . . 3 (𝑋𝐴 → (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ ∃𝑥 𝑥 ∈ {𝑋}))
71, 6mpbird 166 . 2 (𝑋𝐴 → ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}))
8 xpima2m 4893 . 2 (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}) → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
97, 8syl 14 1 (𝑋𝐴 → ((𝐴 × 𝐵) “ {𝑋}) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1290  wex 1427  wcel 1439  cin 3001  wss 3002  {csn 3452   × cxp 4452  cima 4457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-xp 4460  df-rel 4461  df-cnv 4462  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467
This theorem is referenced by: (None)
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