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Theorem feu 6549
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 6509 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fneu2 6457 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → ∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹)
31, 2sylan 582 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹)
4 opelf 6534 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → (𝐶𝐴𝑦𝐵))
54simprd 498 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → 𝑦𝐵)
65ex 415 . . . . . 6 (𝐹:𝐴𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹𝑦𝐵))
76pm4.71rd 565 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ (𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
87eubidv 2668 . . . 4 (𝐹:𝐴𝐵 → (∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
98adantr 483 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
103, 9mpbid 234 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
11 df-reu 3145 . 2 (∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
1210, 11sylibr 236 1 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2110  ∃!weu 2649  ∃!wreu 3140  cop 4567   Fn wfn 6345  wf 6346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-fun 6352  df-fn 6353  df-f 6354
This theorem is referenced by:  fsn  6892  f1ofveu  7145
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