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Theorem funfv2 6996
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 6995 . 2 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
2 funrel 6582 . . . 4 (Fun 𝐹 → Rel 𝐹)
3 relimasn 6102 . . . 4 (Rel 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
42, 3syl 17 . . 3 (Fun 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
54unieqd 4919 . 2 (Fun 𝐹 (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
61, 5eqtrd 2776 1 (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  {cab 2713  {csn 4625   cuni 4906   class class class wbr 5142  cima 5687  Rel wrel 5689  Fun wfun 6554  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568
This theorem is referenced by:  funfv2f  6997
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