Theorem elunirnALT 6139

Description: Alternate proof of elunirn 6138. It is shorter but requires ax-pow 4615 (through eluniima 6137, funiunfv 6135, ndmfv 5872). (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 5335	. . . 4 |- ( F " dom F ) = ran F
2	1	unieqi 4244	. . 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 6137	. 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 1823   E.wrex 2805   U.cuni 4235   dom cdm 4988   ran crn 4989   "cima 4991   Fun wfun 5564   ` cfv 5570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578

This theorem is referenced by: (None)