Theorem funiun 6412

Description: A function is a union of singletons of ordered pairs indexed by its domain. (Contributed by AV, 18-Sep-2020.)

Assertion

Ref	Expression
funiun	|- ( Fun F -> F = U_ x e. dom F { <. x , ( F ` x ) >. } )

Distinct variable group:   x, F

Proof of Theorem funiun

Step	Hyp	Ref	Expression
1		funfn 5918	. . 3 |- ( Fun F <-> F Fn dom F )
2		dffn5 6241	. . 3 |- ( F Fn dom F <-> F = ( x e. dom F |-> ( F ` x ) ) )
3	1, 2	sylbb 209	. 2 |- ( Fun F -> F = ( x e. dom F |-> ( F ` x ) ) )
4		fvex 6201	. . 3 |- ( F ` x ) e. _V
5	4	dfmpt 6410	. 2 |- ( x e. dom F |-> ( F ` x ) ) = U_ x e. dom F { <. x , ( F ` x ) >. }
6	3, 5	syl6eq 2672	1 |- ( Fun F -> F = U_ x e. dom F { <. x , ( F ` x ) >. } )

Syntax hints:   -> wi 4   = wceq 1483   {csn 4177   <.cop 4183   U_ciun 4520   |-> cmpt 4729   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  funopsn  6413