Theorem rnxpid 4775

Description: The range of a square cross product. (Contributed by FL, 17-May-2010.)

Assertion

Ref	Expression
rnxpid	⊢ ran (𝐴 × 𝐴) = 𝐴

Proof of Theorem rnxpid

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnxpss 4774	. 2 ⊢ ran (𝐴 × 𝐴) ⊆ 𝐴
2		opelxp 4392	. . . . . 6 ⊢ (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
3		anidm 388	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
4	2, 3	bitri 182	. . . . 5 ⊢ (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴)
5		opeq1 3570	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑦, 𝑥⟩)
6	5	eleq1d 2147	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
7	6	equcoms 1634	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
8	7	biimpd 142	. . . . . 6 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) → ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
9	8	spimev 1782	. . . . 5 ⊢ (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) → ∃𝑦⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
10	4, 9	sylbir 133	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
11		vex 2604	. . . . 5 ⊢ 𝑥 ∈ V
12	11	elrn2 4594	. . . 4 ⊢ (𝑥 ∈ ran (𝐴 × 𝐴) ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
13	10, 12	sylibr 132	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (𝐴 × 𝐴))
14	13	ssriv 3003	. 2 ⊢ 𝐴 ⊆ ran (𝐴 × 𝐴)
15	1, 14	eqssi 3015	1 ⊢ ran (𝐴 × 𝐴) = 𝐴

Syntax hints:   ∧ wa 102   ↔ wb 103   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ⟨cop 3401   × cxp 4361  ran crn 4364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by: (None)