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Theorem xpider 7763
Description: A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by FL, 31-Jul-2009.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
xpider (𝐴 × 𝐴) Er 𝐴

Proof of Theorem xpider
StepHypRef Expression
1 relxp 5188 . 2 Rel (𝐴 × 𝐴)
2 dmxpid 5305 . 2 dom (𝐴 × 𝐴) = 𝐴
3 cnvxp 5510 . . 3 (𝐴 × 𝐴) = (𝐴 × 𝐴)
4 xpidtr 5477 . . 3 ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
5 uneq1 3738 . . . 4 ((𝐴 × 𝐴) = (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
6 unss2 3762 . . . 4 (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
7 unidm 3734 . . . . 5 ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
8 eqtr 2640 . . . . . 6 ((((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) ∧ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) → ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
9 sseq2 3606 . . . . . . 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)))
109biimpd 219 . . . . . 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)))
118, 10syl 17 . . . . 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) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
127, 11mpan2 706 . . . 4 (((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)))
135, 6, 12syl2im 40 . . 3 ((𝐴 × 𝐴) = (𝐴 × 𝐴) → (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
143, 4, 13mp2 9 . 2 ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)
15 df-er 7687 . 2 ((𝐴 × 𝐴) Er 𝐴 ↔ (Rel (𝐴 × 𝐴) ∧ dom (𝐴 × 𝐴) = 𝐴 ∧ ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
161, 2, 14, 15mpbir3an 1242 1 (𝐴 × 𝐴) Er 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  cun 3553  wss 3555   × cxp 5072  ccnv 5073  dom cdm 5074  ccom 5078  Rel wrel 5079   Er wer 7684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-er 7687
This theorem is referenced by:  riiner  7765  efglem  18050
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