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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.
StepHypRef Expression
1 sneq 3744 . . . 4 (a = A → {a} = {A})
21feq2d 5215 . . 3 (a = A → (F:{a}–→{b} ↔ F:{A}–→{b}))
3 opeq1 4578 . . . . 5 (a = Aa, b = A, b)
43sneqd 3746 . . . 4 (a = A → {a, b} = {A, b})
54eqeq2d 2364 . . 3 (a = A → (F = {a, b} ↔ F = {A, b}))
62, 5bibi12d 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}))
97, 8syl 15 . . 3 (b = B → (F:{A}–→{b} ↔ F:{A}–→{B}))
10 opeq2 4579 . . . . 5 (b = BA, b = A, B)
1110sneqd 3746 . . . 4 (b = B → {A, b} = {A, B})
1211eqeq2d 2364 . . 3 (b = B → (F = {A, b} ↔ F = {A, B}))
139, 12bibi12d 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
1614, 15fsn 5432 . 2 (F:{a}–→{b} ↔ F = {a, b})
176, 13, 16vtocl2g 2918 1 ((A C B D) → (F:{A}–→{B} ↔ F = {A, B}))
 Colors of variables: wff setvar class 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)
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