Theorem rnxpid 5173

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 5170	. 2 ⊢ ran (𝐴 × 𝐴) ⊆ 𝐴
2		opelxp 4757	. . . . . 6 ⊢ (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
3		anidm 396	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
4	2, 3	bitri 184	. . . . 5 ⊢ (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴)
5		opeq1 3863	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑦, 𝑥⟩)
6	5	eleq1d 2299	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
7	6	equcoms 1755	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
8	7	biimpd 144	. . . . . 6 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) → ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
9	8	spimev 1908	. . . . 5 ⊢ (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) → ∃𝑦⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
10	4, 9	sylbir 135	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
11		vex 2804	. . . . 5 ⊢ 𝑥 ∈ V
12	11	elrn2 4976	. . . 4 ⊢ (𝑥 ∈ ran (𝐴 × 𝐴) ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
13	10, 12	sylibr 134	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (𝐴 × 𝐴))
14	13	ssriv 3230	. 2 ⊢ 𝐴 ⊆ ran (𝐴 × 𝐴)
15	1, 14	eqssi 3242	1 ⊢ ran (𝐴 × 𝐴) = 𝐴

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1397  ∃wex 1540   ∈ wcel 2201  ⟨cop 3673   × cxp 4725  ran crn 4728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-br 4090  df-opab 4152  df-xp 4733  df-rel 4734  df-cnv 4735  df-dm 4737  df-rn 4738

This theorem is referenced by: (None)