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Theorem feu 6764
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 6714 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 fneu2 6657 . . . 4 ((𝐹 Fn 𝐴𝐶𝐴) → ∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹)
31, 2sylan 580 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹)
4 opelf 6749 . . . . . . . 8 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → (𝐶𝐴𝑦𝐵))
54simprd 496 . . . . . . 7 ((𝐹:𝐴𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹) → 𝑦𝐵)
65ex 413 . . . . . 6 (𝐹:𝐴𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹𝑦𝐵))
76pm4.71rd 563 . . . . 5 (𝐹:𝐴𝐵 → (⟨𝐶, 𝑦⟩ ∈ 𝐹 ↔ (𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
87eubidv 2580 . . . 4 (𝐹:𝐴𝐵 → (∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
98adantr 481 . . 3 ((𝐹:𝐴𝐵𝐶𝐴) → (∃!𝑦𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹)))
103, 9mpbid 231 . 2 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
11 df-reu 3377 . 2 (∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹 ↔ ∃!𝑦(𝑦𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ 𝐹))
1210, 11sylibr 233 1 ((𝐹:𝐴𝐵𝐶𝐴) → ∃!𝑦𝐵𝐶, 𝑦⟩ ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106  ∃!weu 2562  ∃!wreu 3374  cop 4633   Fn wfn 6535  wf 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-fun 6542  df-fn 6543  df-f 6544
This theorem is referenced by:  fsn  7129  f1ofveu  7399
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