Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpider GIF version

Theorem xpider 6511
 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 4659 . 2 Rel (𝐴 × 𝐴)
2 dmxpid 4771 . 2 dom (𝐴 × 𝐴) = 𝐴
3 cnvxp 4968 . . 3 (𝐴 × 𝐴) = (𝐴 × 𝐴)
4 xpidtr 4940 . . 3 ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
5 uneq1 3229 . . . 4 ((𝐴 × 𝐴) = (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
6 unss2 3253 . . . 4 (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
7 unidm 3225 . . . . 5 ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
8 eqtr 2158 . . . . . 6 ((((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) ∧ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) → ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
9 sseq2 3127 . . . . . . 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 143 . . . . . 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 14 . . . . 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 422 . . . 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 38 . . 3 ((𝐴 × 𝐴) = (𝐴 × 𝐴) → (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
143, 4, 13mp2 16 . 2 ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)
15 df-er 6440 . 2 ((𝐴 × 𝐴) Er 𝐴 ↔ (Rel (𝐴 × 𝐴) ∧ dom (𝐴 × 𝐴) = 𝐴 ∧ ((𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
161, 2, 14, 15mpbir3an 1164 1 (𝐴 × 𝐴) Er 𝐴
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∪ cun 3075   ⊆ wss 3077   × cxp 4548  ◡ccnv 4549  dom cdm 4550   ∘ ccom 4554  Rel wrel 4555   Er wer 6437 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4141 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-br 3939  df-opab 3999  df-xp 4556  df-rel 4557  df-cnv 4558  df-co 4559  df-dm 4560  df-er 6440 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator