Theorem f1osng 5473

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 3587	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2		f1oeq2 5422	. . . 4 ⊢ ({𝑎} = {𝐴} → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
3	1, 2	syl 14	. . 3 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
4		opeq1 3758	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
5	4	sneqd 3589	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
6		f1oeq1 5421	. . . 4 ⊢ ({⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩} → ({⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
7	5, 6	syl 14	. . 3 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
8	3, 7	bitrd 187	. 2 ⊢ (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏}))
9		sneq 3587	. . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵})
10		f1oeq3 5423	. . . 4 ⊢ ({𝑏} = {𝐵} → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵}))
11	9, 10	syl 14	. . 3 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵}))
12		opeq2 3759	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
13	12	sneqd 3589	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
14		f1oeq1 5421	. . . 4 ⊢ ({⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩} → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
15	13, 14	syl 14	. . 3 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝐵} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
16	11, 15	bitrd 187	. 2 ⊢ (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}:{𝐴}–1-1-onto→{𝑏} ↔ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}))
17		vex 2729	. . 3 ⊢ 𝑎 ∈ V
18		vex 2729	. . 3 ⊢ 𝑏 ∈ V
19	17, 18	f1osn 5472	. 2 ⊢ {⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏}
20	8, 16, 19	vtocl2g 2790	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  {csn 3576  ⟨cop 3579  –1-1-onto→wf1o 5187

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195

This theorem is referenced by:  f1sng  5474  f1oprg  5476  fsnunf  5685  dif1en  6845  1fv  10074  zfz1isolem1  10753  sumsnf  11350  prodsnf  11533  ennnfonelemhf1o  12346