Theorem fsng 5434

Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by set.mm contributors, 26-Oct-2012.)

Assertion

Ref	Expression
fsng	⊢ ((A ∈ C ∧ B ∈ D) → (F:{A}–→{B} ↔ F = {⟨A, B⟩}))

Proof of Theorem fsng

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	1	feq2d 5216	. . 3 ⊢ (a = A → (F:{a}–→{b} ↔ F:{A}–→{b}))
3		opeq1 4579	. . . . 5 ⊢ (a = A → ⟨a, b⟩ = ⟨A, b⟩)
4	3	sneqd 3747	. . . 4 ⊢ (a = A → {⟨a, b⟩} = {⟨A, b⟩})
5	4	eqeq2d 2364	. . 3 ⊢ (a = A → (F = {⟨a, b⟩} ↔ F = {⟨A, b⟩}))
6	2, 5	bibi12d 312	. 2 ⊢ (a = A → ((F:{a}–→{b} ↔ F = {⟨a, b⟩}) ↔ (F:{A}–→{b} ↔ F = {⟨A, b⟩})))
7		sneq 3745	. . . 4 ⊢ (b = B → {b} = {B})
8		feq3 5213	. . . 4 ⊢ ({b} = {B} → (F:{A}–→{b} ↔ F:{A}–→{B}))
9	7, 8	syl 15	. . 3 ⊢ (b = B → (F:{A}–→{b} ↔ F:{A}–→{B}))
10		opeq2 4580	. . . . 5 ⊢ (b = B → ⟨A, b⟩ = ⟨A, B⟩)
11	10	sneqd 3747	. . . 4 ⊢ (b = B → {⟨A, b⟩} = {⟨A, B⟩})
12	11	eqeq2d 2364	. . 3 ⊢ (b = B → (F = {⟨A, b⟩} ↔ F = {⟨A, B⟩}))
13	9, 12	bibi12d 312	. 2 ⊢ (b = B → ((F:{A}–→{b} ↔ F = {⟨A, b⟩}) ↔ (F:{A}–→{B} ↔ F = {⟨A, B⟩})))
14		vex 2863	. . 3 ⊢ a ∈ V
15		vex 2863	. . 3 ⊢ b ∈ V
16	14, 15	fsn 5433	. 2 ⊢ (F:{a}–→{b} ↔ F = {⟨a, b⟩})
17	6, 13, 16	vtocl2g 2919	1 ⊢ ((A ∈ C ∧ B ∈ D) → (F:{A}–→{B} ↔ F = {⟨A, B⟩}))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {csn 3738  ⟨cop 4562  –→wf 4778

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-xp 4785  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: (None)