Theorem fvelimab 3841

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 1855	. . 3 |- (C e. (F"B) -> C e. V)
2	1	anim2i 333	. 2 |- (((F Fn A /\ B (_ A) /\ C e. (F"B)) -> ((F Fn A /\ B (_ A) /\ C e. V))
3		fvex 3808	. . . . . 6 |- (F` x) e. V
4		eleq1 1571	. . . . . 6 |- ((F` x) = C -> ((F` x) e. V <-> C e. V))
5	3, 4	mpbii 191	. . . . 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 1781	. . 3 |- (E.x e. B (F` x) = C -> C e. V)
8	7	anim2i 333	. 2 |- (((F Fn A /\ B (_ A) /\ E.x e. B (F` x) = C) -> ((F Fn A /\ B (_ A) /\ C e. V))
9		elimag 3470	. . . 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 3833	. . . . 5 |- ((Fun F /\ x e. dom F /\ C e. V) -> ((F` x) = C <-> xFC))
12		fnfun 3660	. . . . . . 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 2107	. . . . . . . . 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 3662	. . . . . . . . . 10 |- (F Fn A -> dom F = A)
18	17	eleq2d 1578	. . . . . . . . 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 202	. . . . . . 7 |- ((F Fn A /\ B (_ A) -> (x e. B -> x e. dom F))
21	20	imp 348	. . . . . 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 861	. . . 4 |- ((((F Fn A /\ B (_ A) /\ C e. V) /\ x e. B) -> ((F` x) = C <-> xFC))
25	24	rexbidva 1698	. . 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 533	. 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 683	1 |- ((F Fn A /\ B (_ A) -> (C e. (F"B) <-> E.x e. B (F` x) = C))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 988   e. wcel 990  E.wrex 1684  Vcvv 1849   (_ wss 2091   class class class wbr 2669  dom cdm 3225  "cima 3228  Fun wfun 3231   Fn wfn 3232  ` cfv 3237

This theorem is referenced by:  ssimaex 3844  pjimai 10221

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-fv 3253

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