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Theorem funfveu 5540
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 2250 . . . . 5 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
21anbi2d 464 . . . 4 (𝑥 = 𝐴 → ((Fun 𝐹𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹𝐴 ∈ dom 𝐹)))
3 breq1 4018 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐹𝑦𝐴𝐹𝑦))
43eubidv 2044 . . . 4 (𝑥 = 𝐴 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝐴𝐹𝑦))
52, 4imbi12d 234 . . 3 (𝑥 = 𝐴 → (((Fun 𝐹𝑥 ∈ dom 𝐹) → ∃!𝑦 𝑥𝐹𝑦) ↔ ((Fun 𝐹𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)))
6 dffun8 5256 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∃!𝑦 𝑥𝐹𝑦))
76simprbi 275 . . . 4 (Fun 𝐹 → ∀𝑥 ∈ dom 𝐹∃!𝑦 𝑥𝐹𝑦)
87r19.21bi 2575 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ∃!𝑦 𝑥𝐹𝑦)
95, 8vtoclg 2809 . 2 (𝐴 ∈ dom 𝐹 → ((Fun 𝐹𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦))
109anabsi7 581 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  ∃!weu 2036  wcel 2158  wral 2465   class class class wbr 4015  dom cdm 4638  Rel wrel 4643  Fun wfun 5222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-id 4305  df-cnv 4646  df-co 4647  df-dm 4648  df-fun 5230
This theorem is referenced by:  funfvex  5544
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