Theorem meacl 40675

Description: The measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
meacl.1	|- ( ph -> M e. Meas )
meacl.2	|- S = dom M
meacl.3	|- ( ph -> A e. S )

Assertion

Ref	Expression
meacl	|- ( ph -> ( M ` A ) e. ( 0 [,] +oo ) )

Proof of Theorem meacl

Step	Hyp	Ref	Expression
1		id 22	. 2 |- ( ph -> ph )
2		meacl.3	. 2 |- ( ph -> A e. S )
3		meacl.1	. . . 4 |- ( ph -> M e. Meas )
4		meacl.2	. . . 4 |- S = dom M
5	3, 4	meaf 40670	. . 3 |- ( ph -> M : S --> ( 0 [,] +oo ) )
6	5	ffvelrnda 6359	. 2 |- ( ( ph /\ A e. S ) -> ( M ` A ) e. ( 0 [,] +oo ) )
7	1, 2, 6	syl2anc 693	1 |- ( ph -> ( M ` A ) e. ( 0 [,] +oo ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   dom cdm 5114   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   [,]cicc 12178  Meascmea 40666

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-rep 4771  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-pw 4160  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-mea 40667

This theorem is referenced by:  meaxrcl  40678  meassle  40680  meaiunlelem  40685  meage0  40692  voncl  40880