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Theorem xpimasn 4957
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 3611 . . 3 (𝑋𝐴 → ∃𝑥 𝑥 ∈ {𝑋})
2 snssi 3634 . . . . . 6 (𝑋𝐴 → {𝑋} ⊆ 𝐴)
3 dfss1 3250 . . . . . 6 ({𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑋}) = {𝑋})
42, 3sylib 121 . . . . 5 (𝑋𝐴 → (𝐴 ∩ {𝑋}) = {𝑋})
54eleq2d 2187 . . . 4 (𝑋𝐴 → (𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ 𝑥 ∈ {𝑋}))
65exbidv 1781 . . 3 (𝑋𝐴 → (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ ∃𝑥 𝑥 ∈ {𝑋}))
71, 6mpbird 166 . 2 (𝑋𝐴 → ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}))
8 xpima2m 4956 . 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 1316  wex 1453  wcel 1465  cin 3040  wss 3041  {csn 3497   × cxp 4507  cima 4512
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  imasnopn  12395
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