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Theorem fsn 5256
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 4983 . . . . . . . 8 ((𝐹:{A}⟶{B} x, y 𝐹) → (x {A} y {B}))
2 elsn 3361 . . . . . . . . 9 (x {A} ↔ x = A)
3 elsn 3361 . . . . . . . . 9 (y {B} ↔ y = B)
42, 3anbi12i 436 . . . . . . . 8 ((x {A} y {B}) ↔ (x = A y = B))
51, 4sylib 127 . . . . . . 7 ((𝐹:{A}⟶{B} x, y 𝐹) → (x = A y = B))
65ex 108 . . . . . 6 (𝐹:{A}⟶{B} → (⟨x, y 𝐹 → (x = A y = B)))
7 fsn.1 . . . . . . . . . 10 A V
87snid 3373 . . . . . . . . 9 A {A}
9 feu 4993 . . . . . . . . 9 ((𝐹:{A}⟶{B} A {A}) → ∃!y {B}⟨A, y 𝐹)
108, 9mpan2 403 . . . . . . . 8 (𝐹:{A}⟶{B} → ∃!y {B}⟨A, y 𝐹)
113anbi1i 434 . . . . . . . . . . 11 ((y {B} A, y 𝐹) ↔ (y = B A, y 𝐹))
12 opeq2 3520 . . . . . . . . . . . . . 14 (y = B → ⟨A, y⟩ = ⟨A, B⟩)
1312eleq1d 2084 . . . . . . . . . . . . 13 (y = B → (⟨A, y 𝐹 ↔ ⟨A, B 𝐹))
1413pm5.32i 430 . . . . . . . . . . . 12 ((y = B A, y 𝐹) ↔ (y = B A, B 𝐹))
15 ancom 253 . . . . . . . . . . . 12 ((⟨A, B 𝐹 y = B) ↔ (y = B A, B 𝐹))
1614, 15bitr4i 176 . . . . . . . . . . 11 ((y = B A, y 𝐹) ↔ (⟨A, B 𝐹 y = B))
1711, 16bitr2i 174 . . . . . . . . . 10 ((⟨A, B 𝐹 y = B) ↔ (y {B} A, y 𝐹))
1817eubii 1887 . . . . . . . . 9 (∃!y(⟨A, B 𝐹 y = B) ↔ ∃!y(y {B} A, y 𝐹))
19 fsn.2 . . . . . . . . . . . 12 B V
2019eueq1 2686 . . . . . . . . . . 11 ∃!y y = B
2120biantru 286 . . . . . . . . . 10 (⟨A, B 𝐹 ↔ (⟨A, B 𝐹 ∃!y y = B))
22 euanv 1935 . . . . . . . . . 10 (∃!y(⟨A, B 𝐹 y = B) ↔ (⟨A, B 𝐹 ∃!y y = B))
2321, 22bitr4i 176 . . . . . . . . 9 (⟨A, B 𝐹∃!y(⟨A, B 𝐹 y = B))
24 df-reu 2287 . . . . . . . . 9 (∃!y {B}⟨A, y 𝐹∃!y(y {B} A, y 𝐹))
2518, 23, 243bitr4i 201 . . . . . . . 8 (⟨A, B 𝐹∃!y {B}⟨A, y 𝐹)
2610, 25sylibr 137 . . . . . . 7 (𝐹:{A}⟶{B} → ⟨A, B 𝐹)
27 opeq12 3521 . . . . . . . 8 ((x = A y = B) → ⟨x, y⟩ = ⟨A, B⟩)
2827eleq1d 2084 . . . . . . 7 ((x = A y = B) → (⟨x, y 𝐹 ↔ ⟨A, B 𝐹))
2926, 28syl5ibrcom 146 . . . . . 6 (𝐹:{A}⟶{B} → ((x = A y = B) → ⟨x, y 𝐹))
306, 29impbid 120 . . . . 5 (𝐹:{A}⟶{B} → (⟨x, y 𝐹 ↔ (x = A y = B)))
31 vex 2534 . . . . . . . 8 x V
32 vex 2534 . . . . . . . 8 y V
3331, 32opex 3936 . . . . . . 7 x, y V
3433elsnc 3369 . . . . . 6 (⟨x, y {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
357, 19opth2 3947 . . . . . 6 (⟨x, y⟩ = ⟨A, B⟩ ↔ (x = A y = B))
3634, 35bitr2i 174 . . . . 5 ((x = A y = B) ↔ ⟨x, y {⟨A, B⟩})
3730, 36syl6bb 185 . . . 4 (𝐹:{A}⟶{B} → (⟨x, y 𝐹 ↔ ⟨x, y {⟨A, B⟩}))
3837alrimivv 1733 . . 3 (𝐹:{A}⟶{B} → xy(⟨x, y 𝐹 ↔ ⟨x, y {⟨A, B⟩}))
39 frel 4971 . . . 4 (𝐹:{A}⟶{B} → Rel 𝐹)
407, 19relsnop 4367 . . . 4 Rel {⟨A, B⟩}
41 eqrel 4352 . . . 4 ((Rel 𝐹 Rel {⟨A, B⟩}) → (𝐹 = {⟨A, B⟩} ↔ xy(⟨x, y 𝐹 ↔ ⟨x, y {⟨A, B⟩})))
4239, 40, 41sylancl 394 . . 3 (𝐹:{A}⟶{B} → (𝐹 = {⟨A, B⟩} ↔ xy(⟨x, y 𝐹 ↔ ⟨x, y {⟨A, B⟩})))
4338, 42mpbird 156 . 2 (𝐹:{A}⟶{B} → 𝐹 = {⟨A, B⟩})
447, 19f1osn 5087 . . . 4 {⟨A, B⟩}:{A}–1-1-onto→{B}
45 f1oeq1 5038 . . . 4 (𝐹 = {⟨A, B⟩} → (𝐹:{A}–1-1-onto→{B} ↔ {⟨A, B⟩}:{A}–1-1-onto→{B}))
4644, 45mpbiri 157 . . 3 (𝐹 = {⟨A, B⟩} → 𝐹:{A}–1-1-onto→{B})
47 f1of 5047 . . 3 (𝐹:{A}–1-1-onto→{B} → 𝐹:{A}⟶{B})
4846, 47syl 14 . 2 (𝐹 = {⟨A, B⟩} → 𝐹:{A}⟶{B})
4943, 48impbii 117 1 (𝐹:{A}⟶{B} ↔ 𝐹 = {⟨A, B⟩})
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wal 1224   = wceq 1226   wcel 1370  ∃!weu 1878  ∃!wreu 2282  Vcvv 2531  {csn 3346  cop 3349  Rel wrel 4273  wf 4821  1-1-ontowf1o 4824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-reu 2287  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832
This theorem is referenced by:  fsng  5257
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