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Theorem xpimasn 5089
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 3722 . . 3 (𝑋𝐴 → ∃𝑥 𝑥 ∈ {𝑋})
2 snssi 3748 . . . . . 6 (𝑋𝐴 → {𝑋} ⊆ 𝐴)
3 dfss1 3351 . . . . . 6 ({𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑋}) = {𝑋})
42, 3sylib 122 . . . . 5 (𝑋𝐴 → (𝐴 ∩ {𝑋}) = {𝑋})
54eleq2d 2257 . . . 4 (𝑋𝐴 → (𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ 𝑥 ∈ {𝑋}))
65exbidv 1835 . . 3 (𝑋𝐴 → (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ ∃𝑥 𝑥 ∈ {𝑋}))
71, 6mpbird 167 . 2 (𝑋𝐴 → ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}))
8 xpima2m 5088 . 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 1363  wex 1502  wcel 2158  cin 3140  wss 3141  {csn 3604   × cxp 4636  cima 4641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-xp 4644  df-rel 4645  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651
This theorem is referenced by:  imasnopn  14070
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