Theorem xpider 8351

Description: A Cartesian square is an equivalence relation (in general, it is 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

Step	Hyp	Ref	Expression
1		relxp 5537	. 2 ⊢ Rel (𝐴 × 𝐴)
2		dmxpid 5764	. 2 ⊢ dom (𝐴 × 𝐴) = 𝐴
3		cnvxp 5981	. . 3 ⊢ ◡(𝐴 × 𝐴) = (𝐴 × 𝐴)
4		xpidtr 5949	. . 3 ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
5		uneq1 4083	. . . 4 ⊢ (◡(𝐴 × 𝐴) = (𝐴 × 𝐴) → (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
6		unss2 4108	. . . 4 ⊢ (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
7		unidm 4079	. . . . 5 ⊢ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
8		eqtr 2818	. . . . . 6 ⊢ (((◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) ∧ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) → (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
9		sseq2 3941	. . . . . . 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)))
10	9	biimpd 232	. . . . . 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)))
11	8, 10	syl 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)))
12	7, 11	mpan2 690	. . . 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)))
13	5, 6, 12	syl2im 40	. . 3 ⊢ (◡(𝐴 × 𝐴) = (𝐴 × 𝐴) → (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
14	3, 4, 13	mp2 9	. 2 ⊢ (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)
15		df-er 8272	. 2 ⊢ ((𝐴 × 𝐴) Er 𝐴 ↔ (Rel (𝐴 × 𝐴) ∧ dom (𝐴 × 𝐴) = 𝐴 ∧ (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (𝐴 × 𝐴)))
16	1, 2, 14, 15	mpbir3an 1338	1 ⊢ (𝐴 × 𝐴) Er 𝐴

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∪ cun 3879   ⊆ wss 3881   × cxp 5517  ^◡ccnv 5518  dom cdm 5519   ∘ ccom 5523  Rel wrel 5524   Er wer 8269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-er 8272

This theorem is referenced by:  riiner  8353  efglem  18834