Theorem rnxpid 5178

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 5175	. 2 ⊢ ran (𝐴 × 𝐴) ⊆ 𝐴
2		opelxp 4761	. . . . . 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 3867	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑦, 𝑥⟩)
6	5	eleq1d 2300	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
7	6	equcoms 1756	. . . . . . 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 1909	. . . . 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 2806	. . . . 5 ⊢ 𝑥 ∈ V
12	11	elrn2 4980	. . . 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 3232	. 2 ⊢ 𝐴 ⊆ ran (𝐴 × 𝐴)
15	1, 14	eqssi 3244	1 ⊢ ran (𝐴 × 𝐴) = 𝐴

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ⟨cop 3676   × cxp 4729  ran crn 4732

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742

This theorem is referenced by: (None)