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Theorem funfv2 6739
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 6738 . 2 (Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
2 funrel 6360 . . . 4 (Fun 𝐹 → Rel 𝐹)
3 relimasn 5939 . . . 4 (Rel 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
42, 3syl 17 . . 3 (Fun 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
54unieqd 4838 . 2 (Fun 𝐹 (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
61, 5eqtrd 2859 1 (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  {cab 2802  {csn 4549   cuni 4824   class class class wbr 5052  cima 5545  Rel wrel 5547  Fun wfun 6337  cfv 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351
This theorem is referenced by:  funfv2f  6740
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