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Theorem feu 5123
Description: There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
Assertion
Ref Expression
feu ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶

Proof of Theorem feu
StepHypRef Expression
1 ffn 5097 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fneu2 5055 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → ∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹)
31, 2sylan 277 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹)
4 opelf 5113 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → (𝐶𝐴𝑦𝐵))
54simprd 112 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → 𝑦𝐵)
65ex 113 . . . . . 6 (𝐹:𝐴𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹𝑦𝐵))
76pm4.71rd 386 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ (𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
87eubidv 1951 . . . 4 (𝐹:𝐴𝐵 → (∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
98adantr 270 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
103, 9mpbid 145 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
11 df-reu 2360 . 2 (∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
1210, 11sylibr 132 1 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wcel 1434  ∃!weu 1943  ∃!wreu 2355  cop 3419   Fn wfn 4947  wf 4948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-fun 4954  df-fn 4955  df-f 4956
This theorem is referenced by:  fsn  5387  f1ofveu  5551
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