Theorem f1osng 5167

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 3386	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
2		f1oeq2 5118	. . . 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 3549	. . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
5	4	sneqd 3388	. . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
6		f1oeq1 5117	. . . 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 3386	. . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵})
10		f1oeq3 5119	. . . 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 3550	. . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
13	12	sneqd 3388	. . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
14		f1oeq1 5117	. . . 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 2560	. . 3 ⊢ 𝑎 ∈ V
18		vex 2560	. . 3 ⊢ 𝑏 ∈ V
19	17, 18	f1osn 5166	. 2 ⊢ {⟨𝑎, 𝑏⟩}:{𝑎}–1-1-onto→{𝑏}
20	8, 16, 19	vtocl2g 2617	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {csn 3375  ⟨cop 3378  –1-1-onto→wf1o 4901

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909

This theorem is referenced by:  f1oprg  5168  fsnunf  5362  dif1en  6337  1fv  8996