Theorem rnxpid 4805

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 4804	. 2 ⊢ ran (𝐴 × 𝐴) ⊆ 𝐴
2		opelxp 4420	. . . . . 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 3590	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑦, 𝑥⟩)
6	5	eleq1d 2151	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
7	6	equcoms 1636	. . . . . . 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 1784	. . . . 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 2613	. . . . 5 ⊢ 𝑥 ∈ V
12	11	elrn2 4624	. . . 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 3012	. 2 ⊢ 𝐴 ⊆ ran (𝐴 × 𝐴)
15	1, 14	eqssi 3024	1 ⊢ ran (𝐴 × 𝐴) = 𝐴

Syntax hints:   ∧ wa 102   ↔ wb 103   = wceq 1285  ∃wex 1422   ∈ wcel 1434  ⟨cop 3419   × cxp 4389  ran crn 4392

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402

This theorem is referenced by: (None)