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Theorem funfveu 5109
Description: A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
funfveu ((Fun 𝐹 A 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 2078 . . . . 5 (x = A → (x dom 𝐹A dom 𝐹))
21anbi2d 440 . . . 4 (x = A → ((Fun 𝐹 x dom 𝐹) ↔ (Fun 𝐹 A dom 𝐹)))
3 breq1 3737 . . . . 5 (x = A → (x𝐹yA𝐹y))
43eubidv 1886 . . . 4 (x = A → (∃!y x𝐹y∃!y A𝐹y))
52, 4imbi12d 223 . . 3 (x = A → (((Fun 𝐹 x dom 𝐹) → ∃!y x𝐹y) ↔ ((Fun 𝐹 A dom 𝐹) → ∃!y A𝐹y)))
6 dffun8 4851 . . . . 5 (Fun 𝐹 ↔ (Rel 𝐹 x dom 𝐹∃!y x𝐹y))
76simprbi 260 . . . 4 (Fun 𝐹x dom 𝐹∃!y x𝐹y)
87r19.21bi 2381 . . 3 ((Fun 𝐹 x dom 𝐹) → ∃!y x𝐹y)
95, 8vtoclg 2586 . 2 (A dom 𝐹 → ((Fun 𝐹 A dom 𝐹) → ∃!y A𝐹y))
109anabsi7 502 1 ((Fun 𝐹 A dom 𝐹) → ∃!y A𝐹y)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1226   wcel 1370  ∃!weu 1878  wral 2280   class class class wbr 3734  dom cdm 4268  Rel wrel 4273  Fun wfun 4819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-id 4000  df-cnv 4276  df-co 4277  df-dm 4278  df-fun 4827
This theorem is referenced by:  funfvex  5113
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