Theorem xpider 8811

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 5698	. 2 ⊢ Rel (𝐴 × 𝐴)
2		dmxpid 5934	. 2 ⊢ dom (𝐴 × 𝐴) = 𝐴
3		cnvxp 6164	. . 3 ⊢ ◡(𝐴 × 𝐴) = (𝐴 × 𝐴)
4		xpidtr 6131	. . 3 ⊢ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴)
5		uneq1 4155	. . . 4 ⊢ (◡(𝐴 × 𝐴) = (𝐴 × 𝐴) → (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
6		unss2 4181	. . . 4 ⊢ (((𝐴 × 𝐴) ∘ (𝐴 × 𝐴)) ⊆ (𝐴 × 𝐴) → (◡(𝐴 × 𝐴) ∪ ((𝐴 × 𝐴) ∘ (𝐴 × 𝐴))) ⊆ (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)))
7		unidm 4151	. . . . 5 ⊢ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)
8		eqtr 2750	. . . . . 6 ⊢ (((◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) ∧ ((𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴)) → (◡(𝐴 × 𝐴) ∪ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
9		sseq2 4006	. . . . . . 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 228	. . . . . 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 689	. . . 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 8729	. 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 394   = wceq 1533   ∪ cun 3945   ⊆ wss 3947   × cxp 5678  ^◡ccnv 5679  dom cdm 5680   ∘ ccom 5684  Rel wrel 5685   Er wer 8726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-er 8729

This theorem is referenced by:  riiner  8813  efglem  19676