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Theorem xpimasn 4769
 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 3486 . . 3 (𝑋𝐴 → ∃𝑥 𝑥 ∈ {𝑋})
2 snssi 3508 . . . . . 6 (𝑋𝐴 → {𝑋} ⊆ 𝐴)
3 dfss1 3141 . . . . . 6 ({𝑋} ⊆ 𝐴 ↔ (𝐴 ∩ {𝑋}) = {𝑋})
42, 3sylib 127 . . . . 5 (𝑋𝐴 → (𝐴 ∩ {𝑋}) = {𝑋})
54eleq2d 2107 . . . 4 (𝑋𝐴 → (𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ 𝑥 ∈ {𝑋}))
65exbidv 1706 . . 3 (𝑋𝐴 → (∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}) ↔ ∃𝑥 𝑥 ∈ {𝑋}))
71, 6mpbird 156 . 2 (𝑋𝐴 → ∃𝑥 𝑥 ∈ (𝐴 ∩ {𝑋}))
8 xpima2m 4768 . 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 1243  ∃wex 1381   ∈ wcel 1393   ∩ cin 2916   ⊆ wss 2917  {csn 3375   × cxp 4343   “ cima 4348 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by: (None)
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