Theorem fvelimab 3760

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 1814	. . 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 3727	. . . . . 6 |- (F` x) e. V
4		eleq1 1532	. . . . . 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 1741	. . 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 3403	. . . 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 3752	. . . . 5 |- ((Fun F /\ x e. dom F /\ C e. V) -> ((F` x) = C <-> xFC))
12		fnfun 3581	. . . . . . 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 2060	. . . . . . . . 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 3583	. . . . . . . . . 10 |- (F Fn A -> dom F = A)
18	17	eleq2d 1539	. . . . . . . . 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 857	. . . 4 |- ((((F Fn A /\ B (_ A) /\ C e. V) /\ x e. B) -> ((F` x) = C <-> xFC))
25	24	rexbidva 1658	. . 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 530	. 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 679	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 955   e. wcel 957  E.wrex 1644  Vcvv 1808   (_ wss 2044   class class class wbr 2615  dom cdm 3166  "cima 3169  Fun wfun 3172   Fn wfn 3173  ` cfv 3178

This theorem is referenced by:  ssimaex 3763  pjima 10060

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194

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