Theorem f1osng 6893

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}⟶1-1onto{𝐵})

Proof of Theorem f1osng

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

Step	Hyp	Ref	Expression
1		sneq 4594	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2	1	f1oeq2d 6845	. . 3 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}⟶1-1onto{𝑏} ↔ {⟨𝑎, 𝑏⟩}:{𝐴}⟶1-1onto{𝑏}))
3		opeq1 4843	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
4	3	sneqd 4596	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
5	4	f1oeq1d 6844	. . 3 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝐴}⟶1-1onto{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}⟶1-1onto{𝑏}))
6	2, 5	bitrd 282	. 2 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}⟶1-1onto{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}⟶1-1onto{𝑏}))
7		sneq 4594	. . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵})
8	7	f1oeq3d 6846	. . 3 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}⟶1-1onto{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}⟶1-1onto{𝐵}))
9		opeq2 4844	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
10	9	sneqd 4596	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
11	10	f1oeq1d 6844	. . 3 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}⟶1-1onto{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}⟶1-1onto{𝐵}))
12	8, 11	bitrd 282	. 2 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}⟶1-1onto{𝑏} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}⟶1-1onto{𝐵}))
13		vex 3458	. . 3 ⊢ 𝑎 ∈ V
14		vex 3458	. . 3 ⊢ 𝑏 ∈ V
15	13, 14	f1osn 6892	. 2 ⊢ {⟨𝑎, 𝑏⟩}:{𝑎}⟶1-1onto{𝑏}
16	6, 12, 15	vtocl2g 3537	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}⟶1-1onto{𝐵})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {csn 4588  ⟨cop 4838  ⟶1-1ontowf1o 6560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568

This theorem is used by:  f1sng  6894  f1oprswap  6896  f1oprg  6897  f1o2sn  7172  fsnunf  7216  fsnex  7317  suppsnop  8226  mapsnd  8945  ralxpmap  8955  en2sn  9100  enfixsn  9136  fseqenlem1  10133  canthp1lem2  10768  s1f1  14787  sumsnf  15941  prodsn  16161  prodsnf  16163  vdwlem8  17198  gsumws1  19066  symg1bas  19638  dprdsn  20285  eupthp1  30850  poimirlem16  38759  poimirlem17  38760  poimirlem19  38762  poimirlem20  38763  mapfzcons  43931  sumsnd  46259  1hegrlfgr  49445