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Theorem funfveu 5507
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 2233 . . . . 5 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
21anbi2d 461 . . . 4 (𝑥 = 𝐴 → ((Fun 𝐹𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹𝐴 ∈ dom 𝐹)))
3 breq1 3990 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
43eubidv 2027 . . . 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 5224 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∃!𝑦 𝑥𝐹𝑦))
76simprbi 273 . . . 4 (Fun 𝐹 → ∀𝑥 ∈ dom 𝐹∃!𝑦 𝑥𝐹𝑦)
87r19.21bi 2558 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ∃!𝑦 𝑥𝐹𝑦)
95, 8vtoclg 2790 . 2 (𝐴 ∈ dom 𝐹 → ((Fun 𝐹𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦))
109anabsi7 576 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  ∃!weu 2019  wcel 2141  wral 2448   class class class wbr 3987  dom cdm 4609  Rel wrel 4614  Fun wfun 5190
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-id 4276  df-cnv 4617  df-co 4618  df-dm 4619  df-fun 5198
This theorem is referenced by:  funfvex  5511
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