Theorem funfv2 3778

Description: The value of a function. Definition of function value in [Enderton] p. 43.

Assertion

Ref	Expression
funfv2	|- (Fun F -> (F` A) = U.{y | AFy})

Distinct variable groups:   y,A   y,F

Proof of Theorem funfv2

Step	Hyp	Ref	Expression
1		funfv 3777	. 2 |- (Fun F -> (F` A) = U.(F"{A}))
2		funrel 3540	. . . 4 |- (Fun F -> Rel F)
3		relimasn 3432	. . . 4 |- (Rel F -> (F"{A}) = {y | AFy})
4	2, 3	syl 10	. . 3 |- (Fun F -> (F"{A}) = {y | AFy})
5	4	unieqd 2517	. 2 |- (Fun F -> U.(F"{A}) = U.{y | AFy})
6	1, 5	eqtrd 1510	1 |- (Fun F -> (F` A) = U.{y | AFy})

Syntax hints:   -> wi 3   = wceq 958  {cab 1466  {csn 2414  U.cuni 2508   class class class wbr 2625  "cima 3180  Rel wrel 3182  Fun wfun 3183  ` cfv 3189

This theorem is referenced by:  funfv2f 3779

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-pow 2749  ax-pr 2786  ax-un 2873

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-uni 2509  df-br 2626  df-opab 2673  df-id 2842  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-fv 3205

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