Theorem fnexALT 6011

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 5207. This version of fnex 5642 uses ax-pow 4098 and ax-un 4355, whereas fnex 5642 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem fnexALT

Step	Hyp	Ref	Expression
1		fnrel 5221	. . . 4 |- ( F Fn A -> Rel F )
2		relssdmrn 5059	. . . 4 |- ( Rel F -> F C_ ( dom F X. ran F ) )
3	1, 2	syl 14	. . 3 |- ( F Fn A -> F C_ ( dom F X. ran F ) )
4	3	adantr 274	. 2 |- ( ( F Fn A /\ A e. B ) -> F C_ ( dom F X. ran F ) )
5		fndm 5222	. . . . 5 |- ( F Fn A -> dom F = A )
6	5	eleq1d 2208	. . . 4 |- ( F Fn A -> ( dom F e. B <-> A e. B ) )
7	6	biimpar 295	. . 3 |- ( ( F Fn A /\ A e. B ) -> dom F e. B )
8		fnfun 5220	. . . . 5 |- ( F Fn A -> Fun F )
9		funimaexg 5207	. . . . 5 |- ( ( Fun F /\ A e. B ) -> ( F " A ) e. _V )
10	8, 9	sylan 281	. . . 4 |- ( ( F Fn A /\ A e. B ) -> ( F " A ) e. _V )
11		imadmrn 4891	. . . . . . 7 |- ( F " dom F ) = ran F
12	5	imaeq2d 4881	. . . . . . 7 |- ( F Fn A -> ( F " dom F ) = ( F " A ) )
13	11, 12	syl5eqr 2186	. . . . . 6 |- ( F Fn A -> ran F = ( F " A ) )
14	13	eleq1d 2208	. . . . 5 |- ( F Fn A -> ( ran F e. _V <-> ( F " A ) e. _V ) )
15	14	biimpar 295	. . . 4 |- ( ( F Fn A /\ ( F " A ) e. _V ) -> ran F e. _V )
16	10, 15	syldan 280	. . 3 |- ( ( F Fn A /\ A e. B ) -> ran F e. _V )
17		xpexg 4653	. . 3 |- ( ( dom F e. B /\ ran F e. _V ) -> ( dom F X. ran F ) e. _V )
18	7, 16, 17	syl2anc 408	. 2 |- ( ( F Fn A /\ A e. B ) -> ( dom F X. ran F ) e. _V )
19		ssexg 4067	. 2 |- ( ( F C_ ( dom F X. ran F ) /\ ( dom F X. ran F ) e. _V ) -> F e. _V )
20	4, 18, 19	syl2anc 408	1 |- ( ( F Fn A /\ A e. B ) -> F e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   _Vcvv 2686   C_ wss 3071   X. cxp 4537   dom cdm 4539   ran crn 4540   "cima 4542   Rel wrel 4544   Fun wfun 5117   Fn wfn 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fun 5125  df-fn 5126

This theorem is referenced by: (None)