Theorem esumex 26623

Description: An extended sum is a set by definition. (Contributed by Thierry Arnoux, 5-Sep-2017.)

Assertion

Ref	Expression
esumex	|- Σ* k e. A B e. _V

Proof of Theorem esumex

Step	Hyp	Ref	Expression
1		df-esum 26622	. 2 |- Σ* k e. A B = U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) )
2		ovex 6218	. . 3 |- ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) e. _V
3	2	uniex 6479	. 2 |- U. ( ( RR*s ↾s ( 0 [,] +oo ) ) tsums ( k e. A |-> B ) ) e. _V
4	1, 3	eqeltri 2535	1 |- Σ* k e. A B e. _V

Syntax hints:   e. wcel 1758   _Vcvv 3071   U.cuni 4192   |-> cmpt 4451  (class class class)co 6193   0cc0 9386   +oocpnf 9519   [,]cicc 11407   ↾_s cress 14286   RR*scxrs 14549   tsums ctsu 19821  Σ^*cesum 26621

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 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-un 6475

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-sn 3979  df-pr 3981  df-uni 4193  df-iota 5482  df-fv 5527  df-ov 6196  df-esum 26622

This theorem is referenced by:  esumcvg  26673