Theorem f1o2sn 7162

Description: A singleton consisting in a nested ordered pair is a one-to-one function from the cartesian product of two singletons onto a singleton (case where the two singletons are equal). (Contributed by AV, 15-Aug-2019.)

Assertion

Ref	Expression
f1o2sn	⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋})

Proof of Theorem f1o2sn

Step	Hyp	Ref	Expression
1		opex 5469	. . 3 ⊢ ⟨𝐸, 𝐸⟩ ∈ V
2		simpr 484	. . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → 𝑋 ∈ 𝑊)
3		f1osng 6889	. . 3 ⊢ ((⟨𝐸, 𝐸⟩ ∈ V ∧ 𝑋 ∈ 𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
4	1, 2, 3	sylancr 587	. 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
5		xpsng 7159	. . . . . 6 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
6	5	anidms 566	. . . . 5 ⊢ (𝐸 ∈ 𝑉 → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
7	6	eqcomd 2743	. . . 4 ⊢ (𝐸 ∈ 𝑉 → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
8	7	adantr 480	. . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
9	8	f1oeq2d 6844	. 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → ({⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋} ↔ {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋}))
10	4, 9	mpbid 232	1 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3480  {csn 4626  ⟨cop 4632   × cxp 5683  –1-1-onto→wf1o 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568

This theorem is referenced by:  mat1dimelbas  22477