Theorem f1osng 5110

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 3378	. . . 4 ⊢ (𝑎 = A → {𝑎} = {A})
2		f1oeq2 5061	. . . 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 3540	. . . . 5 ⊢ (𝑎 = A → ⟨𝑎, 𝑏⟩ = ⟨A, 𝑏⟩)
5	4	sneqd 3380	. . . 4 ⊢ (𝑎 = A → {⟨𝑎, 𝑏⟩} = {⟨A, 𝑏⟩})
6		f1oeq1 5060	. . . 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 177	. 2 ⊢ (𝑎 = A → ({⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏} ↔ {⟨A, 𝑏⟩}:{A}–1-1-onto→{𝑏}))
9		sneq 3378	. . . 4 ⊢ (𝑏 = B → {𝑏} = {B})
10		f1oeq3 5062	. . . 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 3541	. . . . 5 ⊢ (𝑏 = B → ⟨A, 𝑏⟩ = ⟨A, B⟩)
13	12	sneqd 3380	. . . 4 ⊢ (𝑏 = B → {⟨A, 𝑏⟩} = {⟨A, B⟩})
14		f1oeq1 5060	. . . 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 177	. 2 ⊢ (𝑏 = B → ({⟨A, 𝑏⟩}:{A}–1-1-onto→{𝑏} ↔ {⟨A, B⟩}:{A}–1-1-onto→{B}))
17		vex 2554	. . 3 ⊢ 𝑎 ∈ V
18		vex 2554	. . 3 ⊢ 𝑏 ∈ V
19	17, 18	f1osn 5109	. 2 ⊢ {⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏}
20	8, 16, 19	vtocl2g 2611	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → {⟨A, B⟩}:{A}–1-1-onto→{B})

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {csn 3367  ⟨cop 3370  –1-1-onto→wf1o 4844

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852

This theorem is referenced by:  f1oprg  5111  fsnunf  5305  1fv  8766