Theorem funfvima3 3845

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 2059	. . . . . 6 |- (F (_ G -> (<.A, (F` A)>. e. F -> <.A, (F` A)>. e. G))
2		funfvop 3794	. . . . . 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 2413	. . . . . . . 8 |- (x = A -> {x} = {A})
6	5	imaeq2d 3396	. . . . . . 7 |- (x = A -> (G"{x}) = (G"{A}))
7	6	eleq2d 1538	. . . . . 6 |- (x = A -> ((F` A) e. (G"{x}) <-> (F` A) e. (G"{A})))
8		opeq1 2483	. . . . . . 7 |- (x = A -> <.x, (F` A)>. = <.A, (F` A)>.)
9	8	eleq1d 1537	. . . . . 6 |- (x = A -> (<.x, (F` A)>. e. G <-> <.A, (F` A)>. e. G))
10		visset 1809	. . . . . . 7 |- x e. V
11		fvex 3723	. . . . . . 7 |- (F` A) e. V
12	10, 11	elimasn 3418	. . . . . 6 |- ((F` A) e. (G"{x}) <-> <.x, (F` A)>. e. G)
13	7, 9, 12	vtoclbg 1844	. . . . 5 |- (A e. dom F -> ((F` A) e. (G"{A}) <-> <.A, (F` A)>. e. G))
14	13	ad2antll 407	. . . 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 377	. 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 954   e. wcel 956   (_ wss 2043  {csn 2405  <.cop 2407  dom cdm 3165  "cima 3168  Fun wfun 3171  ` cfv 3177

This theorem is referenced by:  aceq3 4713

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193

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