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Theorem xpiderm 6296
Description: A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by Jim Kingdon, 22-Aug-2019.)
Assertion
Ref Expression
xpiderm (∃𝑥 𝑥𝐴 → (𝐴 × 𝐴) Er 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem xpiderm
StepHypRef Expression
1 relxp 4508 . . 3 Rel (𝐴 × 𝐴)
21a1i 9 . 2 (∃𝑥 𝑥𝐴 → Rel (𝐴 × 𝐴))
3 dmxpm 4617 . 2 (∃𝑥 𝑥𝐴 → dom (𝐴 × 𝐴) = 𝐴)
4 cnvxp 4807 . . . 4 (𝐴 × 𝐴) = (𝐴 × 𝐴)
5 xpidtr 4780 . . . 4 ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
6 uneq1 3133 . . . . 5 ((𝐴 × 𝐴) = (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
7 unss2 3157 . . . . 5 (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
8 unidm 3129 . . . . . 6 ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
9 eqtr 2102 . . . . . . 7 ((((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) ∧ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) → ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
10 sseq2 3034 . . . . . . . 8 (((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴) → (((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) ↔ ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
1110biimpd 142 . . . . . . 7 (((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴) → (((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) → ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
129, 11syl 14 . . . . . 6 ((((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) ∧ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) → (((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) → ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
138, 12mpan2 416 . . . . 5 (((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) → (((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) → ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
146, 7, 13syl2im 38 . . . 4 ((𝐴 × 𝐴) = (𝐴 × 𝐴) → (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
154, 5, 14mp2 16 . . 3 ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)
1615a1i 9 . 2 (∃𝑥 𝑥𝐴 → ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴))
17 df-er 6225 . 2 ((𝐴 × 𝐴) Er 𝐴 ↔ (Rel (𝐴 × 𝐴) ∧ dom (𝐴 × 𝐴) = 𝐴 ∧ ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
182, 3, 16, 17syl3anbrc 1125 1 (∃𝑥 𝑥𝐴 → (𝐴 × 𝐴) Er 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wex 1424  wcel 1436  cun 2984  wss 2986   × cxp 4402  ccnv 4403  dom cdm 4404  ccom 4408  Rel wrel 4409   Er wer 6222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-opab 3869  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-er 6225
This theorem is referenced by: (None)
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