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Theorem xpider 8101
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 5373 . 2 Rel (𝐴 × 𝐴)
2 dmxpid 5590 . 2 dom (𝐴 × 𝐴) = 𝐴
3 cnvxp 5805 . . 3 (𝐴 × 𝐴) = (𝐴 × 𝐴)
4 xpidtr 5773 . . 3 ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
5 uneq1 3983 . . . 4 ((𝐴 × 𝐴) = (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
6 unss2 4007 . . . 4 (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
7 unidm 3979 . . . . 5 ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
8 eqtr 2799 . . . . . 6 ((((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) ∧ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) → ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
9 sseq2 3846 . . . . . . 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 221 . . . . . 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 681 . . . 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 8026 . 2 ((𝐴 × 𝐴) Er 𝐴 ↔ (Rel (𝐴 × 𝐴) ∧ dom (𝐴 × 𝐴) = 𝐴 ∧ ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
161, 2, 14, 15mpbir3an 1398 1 (𝐴 × 𝐴) Er 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  cun 3790  wss 3792   × cxp 5353  ccnv 5354  dom cdm 5355  ccom 5359  Rel wrel 5360   Er wer 8023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-er 8026
This theorem is referenced by:  riiner  8103  efglem  18513
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