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Theorem funfv2 6416
Description: The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)
Assertion
Ref Expression
funfv2 (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹

Proof of Theorem funfv2
StepHypRef Expression
1 funfv 6415 . 2 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
2 funrel 6054 . . . 4 (Fun 𝐹 → Rel 𝐹)
3 relimasn 5634 . . . 4 (Rel 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
42, 3syl 17 . . 3 (Fun 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
54unieqd 4586 . 2 (Fun 𝐹 (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
61, 5eqtrd 2782 1 (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1620  {cab 2734  {csn 4309   cuni 4576   class class class wbr 4792  cima 5257  Rel wrel 5259  Fun wfun 6031  cfv 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-fv 6045
This theorem is referenced by:  funfv2f  6417
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