Theorem funfvima3 2906

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 1502	. . . . . . 7 |- (F (_ G -> (<.A, (F` A)>. e. F -> <.A, (F` A)>. e. G))
2		funopfv 2886	. . . . . . 7 |- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)
3	1, 2	syl5 22	. . . . . 6 |- (F (_ G -> ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. G))
4	3	imp 277	. . . . 5 |- ((F (_ G /\ (Fun F /\ A e. dom F)) -> <.A, (F` A)>. e. G)
5		sneq 1816	. . . . . . . . 9 |- (x = A -> {x} = {A})
6		imaeq2 2603	. . . . . . . . 9 |- ({x} = {A} -> (G"{x}) = (G"{A}))
7	5, 6	syl 12	. . . . . . . 8 |- (x = A -> (G"{x}) = (G"{A}))
8	7	eleq2d 1156	. . . . . . 7 |- (x = A -> ((F` A) e. (G"{x}) <-> (F` A) e. (G"{A})))
9		opeq1 1876	. . . . . . . 8 |- (x = A -> <.x, (F` A)>. = <.A, (F` A)>.)
10	9	eleq1d 1155	. . . . . . 7 |- (x = A -> (<.x, (F` A)>. e. G <-> <.A, (F` A)>. e. G))
11		visset 1350	. . . . . . . 8 |- x e. V
12		fvex 2838	. . . . . . . 8 |- (F` A) e. V
13	11, 12	elimasn 2617	. . . . . . 7 |- ((F` A) e. (G"{x}) <-> <.x, (F` A)>. e. G)
14	8, 10, 13	vtoclbg 1384	. . . . . 6 |- (A e. dom F -> ((F` A) e. (G"{A}) <-> <.A, (F` A)>. e. G))
15	14	ad2antrr 323	. . . . 5 |- ((F (_ G /\ (Fun F /\ A e. dom F)) -> ((F` A) e. (G"{A}) <-> <.A, (F` A)>. e. G))
16	4, 15	mpbird 171	. . . 4 |- ((F (_ G /\ (Fun F /\ A e. dom F)) -> (F` A) e. (G"{A}))
17	16	exp32 294	. . 3 |- (F (_ G -> (Fun F -> (A e. dom F -> (F` A) e. (G"{A}))))
18	17	com12 13	. 2 |- (Fun F -> (F (_ G -> (A e. dom F -> (F` A) e. (G"{A}))))
19	18	imp 277	1 |- ((Fun F /\ F (_ G) -> (A e. dom F -> (F` A) e. (G"{A})))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092   (_ wss 1487  {csn 1808  <.cop 1810  dom cdm 2410  "cima 2413  Fun wfun 2416  ` cfv 2422

This theorem is referenced by:  aceq3 3556

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

metamath.org