Theorem f1osng 6816

Description: A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)

Assertion

Ref	Expression
f1osng	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})

Proof of Theorem f1osng

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4590	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	f1oeq2d 6770	. . 3 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
3		opeq1 4829	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
4	3	sneqd 4592	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
5	4	f1oeq1d 6769	. . 3 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
6	2, 5	bitrd 279	. 2 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
7		sneq 4590	. . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵})
8	7	f1oeq3d 6771	. . 3 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵}))
9		opeq2 4830	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
10	9	sneqd 4592	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
11	10	f1oeq1d 6769	. . 3 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
12	8, 11	bitrd 279	. 2 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
13		vex 3444	. . 3 ⊢ 𝑎 ∈ V
14		vex 3444	. . 3 ⊢ 𝑏 ∈ V
15	13, 14	f1osn 6815	. 2 ⊢ {⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏}
16	6, 12, 15	vtocl2g 3529	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {csn 4580  ⟨cop 4586  –1-1-onto→wf1o 6491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499

This theorem is referenced by:  f1sng  6817  f1oprswap  6819  f1oprg  6820  f1o2sn  7087  fsnunf  7131  fsnex  7229  suppsnop  8120  mapsnd  8824  ralxpmap  8834  en2sn  8978  enfixsn  9014  fseqenlem1  9934  canthp1lem2  10564  sumsnf  15666  prodsn  15885  prodsnf  15887  vdwlem8  16916  gsumws1  18763  symg1bas  19320  dprdsn  19967  eupthp1  30291  s1f1  33025  poimirlem16  37837  poimirlem17  37838  poimirlem19  37840  poimirlem20  37841  mapfzcons  42958  sumsnd  45271  1hegrlfgr  48378