Theorem fsng 5433

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 3744	. . . 4 ⊢ (a = A → {a} = {A})
2	1	feq2d 5215	. . 3 ⊢ (a = A → (F:{a}–→{b} ↔ F:{A}–→{b}))
3		opeq1 4578	. . . . 5 ⊢ (a = A → ⟨a, b⟩ = ⟨A, b⟩)
4	3	sneqd 3746	. . . 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 3744	. . . 4 ⊢ (b = B → {b} = {B})
8		feq3 5212	. . . 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 4579	. . . . 5 ⊢ (b = B → ⟨A, b⟩ = ⟨A, B⟩)
11	10	sneqd 3746	. . . 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 2862	. . 3 ⊢ a ∈ V
15		vex 2862	. . 3 ⊢ b ∈ V
16	14, 15	fsn 5432	. 2 ⊢ (F:{a}–→{b} ↔ F = {⟨a, b⟩})
17	6, 13, 16	vtocl2g 2918	1 ⊢ ((A ∈ C ∧ B ∈ D) → (F:{A}–→{B} ↔ F = {⟨A, B⟩}))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {csn 3737  ⟨cop 4561  –→wf 4777

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794

This theorem is referenced by: (None)