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Theorem funfv2 6233
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 6232 . 2 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
2 funrel 5874 . . . 4 (Fun 𝐹 → Rel 𝐹)
3 relimasn 5457 . . . 4 (Rel 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
42, 3syl 17 . . 3 (Fun 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
54unieqd 4419 . 2 (Fun 𝐹 (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
61, 5eqtrd 2655 1 (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  {cab 2607  {csn 4155   cuni 4409   class class class wbr 4623  cima 5087  Rel wrel 5089  Fun wfun 5851  cfv 5857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865
This theorem is referenced by:  funfv2f  6234
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