Theorem elunirnALT 6077

Description: Alternate proof of elunirn 6076. It is shorter but requires ax-pow 4577 (through eluniima 6075, funiunfv 6073, ndmfv 5822). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
elunirnALT	|- ( Fun F -> ( A e. U. ran F <-> E. x e. dom F A e. ( F ` x ) ) )

Distinct variable groups:   x, A   x, F

Proof of Theorem elunirnALT

Step	Hyp	Ref	Expression
1		imadmrn 5286	. . . 4 |- ( F " dom F ) = ran F
2	1	unieqi 4207	. . 3 |- U. ( F " dom F ) = U. ran F
3	2	eleq2i 2532	. 2 |- ( A e. U. ( F " dom F ) <-> A e. U. ran F )
4		eluniima 6075	. 2 |- ( Fun F -> ( A e. U. ( F " dom F ) <-> E. x e. dom F A e. ( F ` x ) ) )
5	3, 4	syl5bbr 259	1 |- ( Fun F -> ( A e. U. ran F <-> E. x e. dom F A e. ( F ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 184   e. wcel 1758   E.wrex 2799   U.cuni 4198   dom cdm 4947   ran crn 4948   "cima 4950   Fun wfun 5519   ` cfv 5525

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-fv 5533

This theorem is referenced by: (None)