Theorem fvelimab 3750

Description: Function value in an image.

Assertion

Ref	Expression
fvelimab	|- ((F Fn A /\ B (_ A) -> (C e. (F"B) <-> E.x e. B (F` x) = C))

Distinct variable groups:   x,A   x,B   x,F   x,C

Proof of Theorem fvelimab

Step	Hyp	Ref	Expression
1		elisset 1808	. . 3 |- (C e. (F"B) -> C e. V)
2	1	anim2i 335	. 2 |- (((F Fn A /\ B (_ A) /\ C e. (F"B)) -> ((F Fn A /\ B (_ A) /\ C e. V))
3		fvex 3717	. . . . . 6 |- (F` x) e. V
4		eleq1 1526	. . . . . 6 |- ((F` x) = C -> ((F` x) e. V <-> C e. V))
5	3, 4	mpbii 193	. . . . 5 |- ((F` x) = C -> C e. V)
6	5	a1i 8	. . . 4 |- (x e. B -> ((F` x) = C -> C e. V))
7	6	r19.23aiv 1735	. . 3 |- (E.x e. B (F` x) = C -> C e. V)
8	7	anim2i 335	. 2 |- (((F Fn A /\ B (_ A) /\ E.x e. B (F` x) = C) -> ((F Fn A /\ B (_ A) /\ C e. V))
9		elimag 3391	. . . 4 |- (C e. V -> (C e. (F"B) <-> E.x e. B xFC))
10	9	adantl 388	. . 3 |- (((F Fn A /\ B (_ A) /\ C e. V) -> (C e. (F"B) <-> E.x e. B xFC))
11		funbrfvbg 3742	. . . . 5 |- ((Fun F /\ x e. dom F /\ C e. V) -> ((F` x) = C <-> xFC))
12		fnfun 3571	. . . . . . 7 |- (F Fn A -> Fun F)
13	12	adantr 389	. . . . . 6 |- ((F Fn A /\ B (_ A) -> Fun F)
14	13	ad2antrr 404	. . . . 5 |- ((((F Fn A /\ B (_ A) /\ C e. V) /\ x e. B) -> Fun F)
15		ssel 2053	. . . . . . . . 9 |- (B (_ A -> (x e. B -> x e. A))
16	15	adantl 388	. . . . . . . 8 |- ((F Fn A /\ B (_ A) -> (x e. B -> x e. A))
17		fndm 3573	. . . . . . . . . 10 |- (F Fn A -> dom F = A)
18	17	eleq2d 1533	. . . . . . . . 9 |- (F Fn A -> (x e. dom F <-> x e. A))
19	18	adantr 389	. . . . . . . 8 |- ((F Fn A /\ B (_ A) -> (x e. dom F <-> x e. A))
20	16, 19	sylibrd 204	. . . . . . 7 |- ((F Fn A /\ B (_ A) -> (x e. B -> x e. dom F))
21	20	imp 350	. . . . . 6 |- (((F Fn A /\ B (_ A) /\ x e. B) -> x e. dom F)
22	21	adantlr 393	. . . . 5 |- ((((F Fn A /\ B (_ A) /\ C e. V) /\ x e. B) -> x e. dom F)
23		simplr 413	. . . . 5 |- ((((F Fn A /\ B (_ A) /\ C e. V) /\ x e. B) -> C e. V)
24	11, 14, 22, 23	syl3anc 856	. . . 4 |- ((((F Fn A /\ B (_ A) /\ C e. V) /\ x e. B) -> ((F` x) = C <-> xFC))
25	24	rexbidva 1652	. . 3 |- (((F Fn A /\ B (_ A) /\ C e. V) -> (E.x e. B (F` x) = C <-> E.x e. B xFC))
26	10, 25	bitr4d 529	. 2 |- (((F Fn A /\ B (_ A) /\ C e. V) -> (C e. (F"B) <-> E.x e. B (F` x) = C))
27	2, 8, 26	pm5.21nd 678	1 |- ((F Fn A /\ B (_ A) -> (C e. (F"B) <-> E.x e. B (F` x) = C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wrex 1638  Vcvv 1802   (_ wss 2037   class class class wbr 2609  dom cdm 3160  "cima 3163  Fun wfun 3166   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  ssimaex 3753  pjima 10015

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

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