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Theorem funsn 5147
 Description: A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.) (Revised by Scott Fenton, 16-Apr-2021.)
Assertion
Ref Expression
funsn Fun {A, B}

Proof of Theorem funsn
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun6 5124 . 2 (Fun {A, B} ↔ x∃*y x{A, B}y)
2 moeq 3012 . . . 4 ∃*y y = B
32a1i 10 . . 3 (x = A∃*y y = B)
4 df-br 4640 . . . . . 6 (x{A, B}yx, y {A, B})
5 vex 2862 . . . . . . . . 9 x V
6 vex 2862 . . . . . . . . 9 y V
75, 6opex 4588 . . . . . . . 8 x, y V
87elsnc 3756 . . . . . . 7 (x, y {A, B} ↔ x, y = A, B)
9 opth 4602 . . . . . . 7 (x, y = A, B ↔ (x = A y = B))
108, 9bitri 240 . . . . . 6 (x, y {A, B} ↔ (x = A y = B))
114, 10bitri 240 . . . . 5 (x{A, B}y ↔ (x = A y = B))
1211mobii 2240 . . . 4 (∃*y x{A, B}y∃*y(x = A y = B))
13 moanimv 2262 . . . 4 (∃*y(x = A y = B) ↔ (x = A∃*y y = B))
1412, 13bitri 240 . . 3 (∃*y x{A, B}y ↔ (x = A∃*y y = B))
153, 14mpbir 200 . 2 ∃*y x{A, B}y
161, 15mpgbir 1550 1 Fun {A, B}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  {csn 3737  ⟨cop 4561   class class class wbr 4639  Fun wfun 4775 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-id 4767  df-cnv 4785  df-fun 4789 This theorem is referenced by:  funsngOLD  5148  funprg  5149  fnsn  5152  fvsn  5445
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