Theorem funfvima3 3860

Description: A class including a function contains the function's value in the image of the singleton of the argument.

Assertion

Ref	Expression
funfvima3	|- ((Fun F /\ F (_ G) -> (A e. dom F -> (F` A) e. (G"{A})))

Proof of Theorem funfvima3

Step	Hyp	Ref	Expression
1		ssel 2066	. . . . . 6 |- (F (_ G -> (<.A, (F` A)>. e. F -> <.A, (F` A)>. e. G))
2		funfvop 3809	. . . . . 6 |- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)
3	1, 2	syl5 21	. . . . 5 |- (F (_ G -> ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. G))
4	3	imp 350	. . . 4 |- ((F (_ G /\ (Fun F /\ A e. dom F)) -> <.A, (F` A)>. e. G)
5		sneq 2421	. . . . . . . 8 |- (x = A -> {x} = {A})
6	5	imaeq2d 3410	. . . . . . 7 |- (x = A -> (G"{x}) = (G"{A}))
7	6	eleq2d 1544	. . . . . 6 |- (x = A -> ((F` A) e. (G"{x}) <-> (F` A) e. (G"{A})))
8		opeq1 2491	. . . . . . 7 |- (x = A -> <.x, (F` A)>. = <.A, (F` A)>.)
9	8	eleq1d 1543	. . . . . 6 |- (x = A -> (<.x, (F` A)>. e. G <-> <.A, (F` A)>. e. G))
10		visset 1816	. . . . . . 7 |- x e. V
11		fvex 3738	. . . . . . 7 |- (F` A) e. V
12	10, 11	elimasn 3432	. . . . . 6 |- ((F` A) e. (G"{x}) <-> <.x, (F` A)>. e. G)
13	7, 9, 12	vtoclbg 1851	. . . . 5 |- (A e. dom F -> ((F` A) e. (G"{A}) <-> <.A, (F` A)>. e. G))
14	13	ad2antll 409	. . . 4 |- ((F (_ G /\ (Fun F /\ A e. dom F)) -> ((F` A) e. (G"{A}) <-> <.A, (F` A)>. e. G))
15	4, 14	mpbird 196	. . 3 |- ((F (_ G /\ (Fun F /\ A e. dom F)) -> (F` A) e. (G"{A}))
16	15	exp32 379	. 2 |- (F (_ G -> (Fun F -> (A e. dom F -> (F` A) e. (G"{A}))))
17	16	impcom 351	1 |- ((Fun F /\ F (_ G) -> (A e. dom F -> (F` A) e. (G"{A})))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960   (_ wss 2050  {csn 2413  <.cop 2415  dom cdm 3176  "cima 3179  Fun wfun 3182  ` cfv 3188

This theorem is referenced by:  aceq3 4743

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 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

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-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

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