Theorem f1osng 5324

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 3745	. . . 4 ⊢ (a = A → {a} = {A})
2		f1oeq2 5283	. . . 4 ⊢ ({a} = {A} → ({⟨a, b⟩}:{a}–1-1-onto→{b} ↔ {⟨a, b⟩}:{A}–1-1-onto→{b}))
3	1, 2	syl 15	. . 3 ⊢ (a = A → ({⟨a, b⟩}:{a}–1-1-onto→{b} ↔ {⟨a, b⟩}:{A}–1-1-onto→{b}))
4		opeq1 4579	. . . 4 ⊢ (a = A → ⟨a, b⟩ = ⟨A, b⟩)
5		sneq 3745	. . . 4 ⊢ (⟨a, b⟩ = ⟨A, b⟩ → {⟨a, b⟩} = {⟨A, b⟩})
6		f1oeq1 5282	. . . 4 ⊢ ({⟨a, b⟩} = {⟨A, b⟩} → ({⟨a, b⟩}:{A}–1-1-onto→{b} ↔ {⟨A, b⟩}:{A}–1-1-onto→{b}))
7	4, 5, 6	3syl 18	. . 3 ⊢ (a = A → ({⟨a, b⟩}:{A}–1-1-onto→{b} ↔ {⟨A, b⟩}:{A}–1-1-onto→{b}))
8	3, 7	bitrd 244	. 2 ⊢ (a = A → ({⟨a, b⟩}:{a}–1-1-onto→{b} ↔ {⟨A, b⟩}:{A}–1-1-onto→{b}))
9		sneq 3745	. . . 4 ⊢ (b = B → {b} = {B})
10		f1oeq3 5284	. . . 4 ⊢ ({b} = {B} → ({⟨A, b⟩}:{A}–1-1-onto→{b} ↔ {⟨A, b⟩}:{A}–1-1-onto→{B}))
11	9, 10	syl 15	. . 3 ⊢ (b = B → ({⟨A, b⟩}:{A}–1-1-onto→{b} ↔ {⟨A, b⟩}:{A}–1-1-onto→{B}))
12		opeq2 4580	. . . 4 ⊢ (b = B → ⟨A, b⟩ = ⟨A, B⟩)
13		sneq 3745	. . . 4 ⊢ (⟨A, b⟩ = ⟨A, B⟩ → {⟨A, b⟩} = {⟨A, B⟩})
14		f1oeq1 5282	. . . 4 ⊢ ({⟨A, b⟩} = {⟨A, B⟩} → ({⟨A, b⟩}:{A}–1-1-onto→{B} ↔ {⟨A, B⟩}:{A}–1-1-onto→{B}))
15	12, 13, 14	3syl 18	. . 3 ⊢ (b = B → ({⟨A, b⟩}:{A}–1-1-onto→{B} ↔ {⟨A, B⟩}:{A}–1-1-onto→{B}))
16	11, 15	bitrd 244	. 2 ⊢ (b = B → ({⟨A, b⟩}:{A}–1-1-onto→{b} ↔ {⟨A, B⟩}:{A}–1-1-onto→{B}))
17		vex 2863	. . 3 ⊢ a ∈ V
18		vex 2863	. . 3 ⊢ b ∈ V
19	17, 18	f1osn 5323	. 2 ⊢ {⟨a, b⟩}:{a}–1-1-onto→{b}
20	8, 16, 19	vtocl2g 2919	1 ⊢ ((A ∈ V ∧ B ∈ W) → {⟨A, B⟩}:{A}–1-1-onto→{B})

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {csn 3738  ⟨cop 4562  –1-1-onto→wf1o 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795

This theorem is referenced by:  en2sn  6048