Theorem fnexALT 4536

Description: If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 4495.

Assertion

Ref	Expression
fnexALT	|- ((F Fn A /\ A e. B) -> F e. _V)

Proof of Theorem fnexALT

Step	Hyp	Ref	Expression
1		fnrel 4511	. . . 4 |- (F Fn A -> Rel F)
2		relssdmrn 4416	. . . 4 |- (Rel F -> F C_ (dom F X. ran F))
3	1, 2	syl 12	. . 3 |- (F Fn A -> F C_ (dom F X. ran F))
4	3	adantr 425	. 2 |- ((F Fn A /\ A e. B) -> F C_ (dom F X. ran F))
5		fndm 4512	. . . . 5 |- (F Fn A -> dom F = A)
6	5	eleq1d 1963	. . . 4 |- (F Fn A -> (dom F e. B <-> A e. B))
7	6	biimpar 461	. . 3 |- ((F Fn A /\ A e. B) -> dom F e. B)
8		funimaexg 4495	. . . . 5 |- ((Fun F /\ A e. B) -> (F"A) e. _V)
9		fnfun 4510	. . . . 5 |- (F Fn A -> Fun F)
10	8, 9	sylan 497	. . . 4 |- ((F Fn A /\ A e. B) -> (F"A) e. _V)
11	5	imaeq2d 4264	. . . . . . 7 |- (F Fn A -> (F"dom F) = (F"A))
12		imadmrn 4277	. . . . . . 7 |- (F"dom F) = ran F
13	11, 12	syl5eqr 1942	. . . . . 6 |- (F Fn A -> ran F = (F"A))
14	13	eleq1d 1963	. . . . 5 |- (F Fn A -> (ran F e. _V <-> (F"A) e. _V))
15	14	biimpar 461	. . . 4 |- ((F Fn A /\ (F"A) e. _V) -> ran F e. _V)
16	10, 15	syldan 516	. . 3 |- ((F Fn A /\ A e. B) -> ran F e. _V)
17		xpexg 4095	. . 3 |- ((dom F e. B /\ ran F e. _V) -> (dom F X. ran F) e. _V)
18	7, 16, 17	syl11anc 524	. 2 |- ((F Fn A /\ A e. B) -> (dom F X. ran F) e. _V)
19		ssexg 3457	. 2 |- ((F C_ (dom F X. ran F) /\ (dom F X. ran F) e. _V) -> F e. _V)
20	4, 18, 19	syl11anc 524	1 |- ((F Fn A /\ A e. B) -> F e. _V)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  _Vcvv 2292   C_ wss 2593   X. cxp 3984  dom cdm 3986  ran crn 3987  "cima 3989  Rel wrel 3991  Fun wfun 3992   Fn wfn 3993

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009

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