Theorem f1o2sn 7118

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 5428	. . 3 ⊢ ⟨𝐸, 𝐸⟩ ∈ V
2		simpr 488	. . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → 𝑋 ∈ 𝑊)
3		f1osng 6843	. . 3 ⊢ ((⟨𝐸, 𝐸⟩ ∈ V ∧ 𝑋 ∈ 𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
4	1, 2, 3	sylancr 596	. 2 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:{⟨𝐸, 𝐸⟩}–1-1-onto→{𝑋})
5		xpsng 7115	. . . . . 6 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
6	5	anidms 574	. . . . 5 ⊢ (𝐸 ∈ 𝑉 → ({𝐸} × {𝐸}) = {⟨𝐸, 𝐸⟩})
7	6	eqcomd 2767	. . . 4 ⊢ (𝐸 ∈ 𝑉 → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
8	7	adantr 484	. . 3 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {⟨𝐸, 𝐸⟩} = ({𝐸} × {𝐸}))
9	8	f1oeq2d 6796	. 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 234	1 ⊢ ((𝐸 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → {⟨⟨𝐸, 𝐸⟩, 𝑋⟩}:({𝐸} × {𝐸})–1-1-onto→{𝑋})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453  {csn 4579  ⟨cop 4585   × cxp 5641  –1-1-onto→wf1o 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522

This theorem is referenced by:  mat1dimelbas  22518