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Theorem fsn 5432
Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 10-Dec-2003.)
Hypotheses
Ref Expression
fsn.1 A V
fsn.2 B V
Assertion
Ref Expression
fsn (F:{A}–→{B} ↔ F = {A, B})

Proof of Theorem fsn
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opelf 5235 . . . . . . 7 ((F:{A}–→{B} x, y F) → (x {A} y {B}))
2 elsn 3748 . . . . . . . 8 (x {A} ↔ x = A)
3 elsn 3748 . . . . . . . 8 (y {B} ↔ y = B)
42, 3anbi12i 678 . . . . . . 7 ((x {A} y {B}) ↔ (x = A y = B))
51, 4sylib 188 . . . . . 6 ((F:{A}–→{B} x, y F) → (x = A y = B))
65ex 423 . . . . 5 (F:{A}–→{B} → (x, y F → (x = A y = B)))
7 fsn.1 . . . . . . . . 9 A V
87snid 3760 . . . . . . . 8 A {A}
9 feu 5242 . . . . . . . 8 ((F:{A}–→{B} A {A}) → ∃!y {B}A, y F)
108, 9mpan2 652 . . . . . . 7 (F:{A}–→{B} → ∃!y {B}A, y F)
113anbi1i 676 . . . . . . . . . 10 ((y {B} A, y F) ↔ (y = B A, y F))
12 opeq2 4579 . . . . . . . . . . . . 13 (y = BA, y = A, B)
1312eleq1d 2419 . . . . . . . . . . . 12 (y = B → (A, y FA, B F))
1413pm5.32i 618 . . . . . . . . . . 11 ((y = B A, y F) ↔ (y = B A, B F))
15 ancom 437 . . . . . . . . . . 11 ((A, B F y = B) ↔ (y = B A, B F))
1614, 15bitr4i 243 . . . . . . . . . 10 ((y = B A, y F) ↔ (A, B F y = B))
1711, 16bitr2i 241 . . . . . . . . 9 ((A, B F y = B) ↔ (y {B} A, y F))
1817eubii 2213 . . . . . . . 8 (∃!y(A, B F y = B) ↔ ∃!y(y {B} A, y F))
19 fsn.2 . . . . . . . . . . 11 B V
2019eueq1 3009 . . . . . . . . . 10 ∃!y y = B
2120biantru 491 . . . . . . . . 9 (A, B F ↔ (A, B F ∃!y y = B))
22 euanv 2265 . . . . . . . . 9 (∃!y(A, B F y = B) ↔ (A, B F ∃!y y = B))
2321, 22bitr4i 243 . . . . . . . 8 (A, B F∃!y(A, B F y = B))
24 df-reu 2621 . . . . . . . 8 (∃!y {B}A, y F∃!y(y {B} A, y F))
2518, 23, 243bitr4i 268 . . . . . . 7 (A, B F∃!y {B}A, y F)
2610, 25sylibr 203 . . . . . 6 (F:{A}–→{B} → A, B F)
27 opeq12 4580 . . . . . . 7 ((x = A y = B) → x, y = A, B)
2827eleq1d 2419 . . . . . 6 ((x = A y = B) → (x, y FA, B F))
2926, 28syl5ibrcom 213 . . . . 5 (F:{A}–→{B} → ((x = A y = B) → x, y F))
306, 29impbid 183 . . . 4 (F:{A}–→{B} → (x, y F ↔ (x = A y = B)))
31 vex 2862 . . . . . . 7 x V
32 vex 2862 . . . . . . 7 y V
3331, 32opex 4588 . . . . . 6 x, y V
3433elsnc 3756 . . . . 5 (x, y {A, B} ↔ x, y = A, B)
35 opth 4602 . . . . 5 (x, y = A, B ↔ (x = A y = B))
3634, 35bitr2i 241 . . . 4 ((x = A y = B) ↔ x, y {A, B})
3730, 36syl6bb 252 . . 3 (F:{A}–→{B} → (x, y Fx, y {A, B}))
3837eqrelrdv 4852 . 2 (F:{A}–→{B} → F = {A, B})
397, 19f1osn 5322 . . . 4 {A, B}:{A}–1-1-onto→{B}
40 f1oeq1 5281 . . . 4 (F = {A, B} → (F:{A}–1-1-onto→{B} ↔ {A, B}:{A}–1-1-onto→{B}))
4139, 40mpbiri 224 . . 3 (F = {A, B} → F:{A}–1-1-onto→{B})
42 f1of 5287 . . 3 (F:{A}–1-1-onto→{B} → F:{A}–→{B})
4341, 42syl 15 . 2 (F = {A, B} → F:{A}–→{B})
4438, 43impbii 180 1 (F:{A}–→{B} ↔ F = {A, B})
Colors of variables: wff setvar class
Syntax hints:  wb 176   wa 358   = wceq 1642   wcel 1710  ∃!weu 2204  ∃!wreu 2616  Vcvv 2859  {csn 3737  cop 4561  –→wf 4777  1-1-ontowf1o 4780
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 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:  fsng  5433  fsn2  5434  xpsn  5435  mapsn  6026
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