Theorem rnxpid 5045

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 5042	. 2 ⊢ ran (𝐴 × 𝐴) ⊆ 𝐴
2		opelxp 4641	. . . . . 6 ⊢ (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
3		anidm 394	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
4	2, 3	bitri 183	. . . . 5 ⊢ (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ 𝑥 ∈ 𝐴)
5		opeq1 3765	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ⟨𝑥, 𝑥⟩ = ⟨𝑦, 𝑥⟩)
6	5	eleq1d 2239	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
7	6	equcoms 1701	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) ↔ ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
8	7	biimpd 143	. . . . . 6 ⊢ (𝑦 = 𝑥 → (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) → ⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴)))
9	8	spimev 1854	. . . . 5 ⊢ (⟨𝑥, 𝑥⟩ ∈ (𝐴 × 𝐴) → ∃𝑦⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
10	4, 9	sylbir 134	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ∃𝑦⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
11		vex 2733	. . . . 5 ⊢ 𝑥 ∈ V
12	11	elrn2 4853	. . . 4 ⊢ (𝑥 ∈ ran (𝐴 × 𝐴) ↔ ∃𝑦⟨𝑦, 𝑥⟩ ∈ (𝐴 × 𝐴))
13	10, 12	sylibr 133	. . 3 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ran (𝐴 × 𝐴))
14	13	ssriv 3151	. 2 ⊢ 𝐴 ⊆ ran (𝐴 × 𝐴)
15	1, 14	eqssi 3163	1 ⊢ ran (𝐴 × 𝐴) = 𝐴

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ⟨cop 3586   × cxp 4609  ran crn 4612

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622

This theorem is referenced by: (None)