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Theorem fsn 5278
 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 5005 . . . . . . . 8 ((𝐹:{A}⟶{B} x, y 𝐹) → (x {A} y {B}))
2 elsn 3382 . . . . . . . . 9 (x {A} ↔ x = A)
3 elsn 3382 . . . . . . . . 9 (y {B} ↔ y = B)
42, 3anbi12i 433 . . . . . . . 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 3394 . . . . . . . . 9 A {A}
9 feu 5015 . . . . . . . . 9 ((𝐹:{A}⟶{B} A {A}) → ∃!y {B}⟨A, y 𝐹)
108, 9mpan2 401 . . . . . . . 8 (𝐹:{A}⟶{B} → ∃!y {B}⟨A, y 𝐹)
113anbi1i 431 . . . . . . . . . . 11 ((y {B} A, y 𝐹) ↔ (y = B A, y 𝐹))
12 opeq2 3541 . . . . . . . . . . . . . 14 (y = B → ⟨A, y⟩ = ⟨A, B⟩)
1312eleq1d 2103 . . . . . . . . . . . . 13 (y = B → (⟨A, y 𝐹 ↔ ⟨A, B 𝐹))
1413pm5.32i 427 . . . . . . . . . . . 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 1906 . . . . . . . . 9 (∃!y(⟨A, B 𝐹 y = B) ↔ ∃!y(y {B} A, y 𝐹))
19 fsn.2 . . . . . . . . . . . 12 B V
2019eueq1 2707 . . . . . . . . . . 11 ∃!y y = B
2120biantru 286 . . . . . . . . . 10 (⟨A, B 𝐹 ↔ (⟨A, B 𝐹 ∃!y y = B))
22 euanv 1954 . . . . . . . . . 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 2307 . . . . . . . . 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 3542 . . . . . . . 8 ((x = A y = B) → ⟨x, y⟩ = ⟨A, B⟩)
2827eleq1d 2103 . . . . . . 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 2554 . . . . . . . 8 x V
32 vex 2554 . . . . . . . 8 y V
3331, 32opex 3957 . . . . . . 7 x, y V
3433elsnc 3390 . . . . . 6 (⟨x, y {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
357, 19opth2 3968 . . . . . 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 1752 . . 3 (𝐹:{A}⟶{B} → xy(⟨x, y 𝐹 ↔ ⟨x, y {⟨A, B⟩}))
39 frel 4992 . . . 4 (𝐹:{A}⟶{B} → Rel 𝐹)
407, 19relsnop 4387 . . . 4 Rel {⟨A, B⟩}
41 eqrel 4372 . . . 4 ((Rel 𝐹 Rel {⟨A, B⟩}) → (𝐹 = {⟨A, B⟩} ↔ xy(⟨x, y 𝐹 ↔ ⟨x, y {⟨A, B⟩})))
4239, 40, 41sylancl 392 . . 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 5109 . . . 4 {⟨A, B⟩}:{A}–1-1-onto→{B}
45 f1oeq1 5060 . . . 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 5069 . . 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 1240   = wceq 1242   ∈ wcel 1390  ∃!weu 1897  ∃!wreu 2302  Vcvv 2551  {csn 3367  ⟨cop 3370  Rel wrel 4293  ⟶wf 4841  –1-1-onto→wf1o 4844 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852 This theorem is referenced by:  fsng  5279
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