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Theorem funfveu 5499
Description: A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
funfveu ((Fun 𝐹𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem funfveu
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2229 . . . . 5 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
21anbi2d 460 . . . 4 (𝑥 = 𝐴 → ((Fun 𝐹𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹𝐴 ∈ dom 𝐹)))
3 breq1 3985 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
43eubidv 2022 . . . 4 (𝑥 = 𝐴 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦))
52, 4imbi12d 233 . . 3 (𝑥 = 𝐴 → (((Fun 𝐹𝑥 ∈ dom 𝐹) → ∃!𝑦 𝑥𝐹𝑦) ↔ ((Fun 𝐹𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)))
6 dffun8 5216 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∃!𝑦 𝑥𝐹𝑦))
76simprbi 273 . . . 4 (Fun 𝐹 → ∀𝑥 ∈ dom 𝐹∃!𝑦 𝑥𝐹𝑦)
87r19.21bi 2554 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ∃!𝑦 𝑥𝐹𝑦)
95, 8vtoclg 2786 . 2 (𝐴 ∈ dom 𝐹 → ((Fun 𝐹𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦))
109anabsi7 571 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1343  ∃!weu 2014  wcel 2136  wral 2444   class class class wbr 3982  dom cdm 4604  Rel wrel 4609  Fun wfun 5182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-cnv 4612  df-co 4613  df-dm 4614  df-fun 5190
This theorem is referenced by:  funfvex  5503
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