Theorem caragenfiiuncl 40729

Description: The Caratheodory's construction is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
caragenfiiuncl.kph	|- F/ k ph
caragenfiiuncl.o	|- ( ph -> O e. OutMeas )
caragenfiiuncl.s	|- S = (CaraGen ` O )
caragenfiiuncl.a	|- ( ph -> A e. Fin )
caragenfiiuncl.b	|- ( ( ph /\ k e. A ) -> B e. S )

Assertion

Ref	Expression
caragenfiiuncl	|- ( ph -> U_ k e. A B e. S )

Distinct variable groups:   A, k   S, k

Allowed substitution hints:   ph( k)   B( k)   O( k)

Proof of Theorem caragenfiiuncl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iuneq1 4534	. . . . 5 |- ( A = (/) -> U_ k e. A B = U_ k e. (/) B )
2		0iun 4577	. . . . . 6 |- U_ k e. (/) B = (/)
3	2	a1i 11	. . . . 5 |- ( A = (/) -> U_ k e. (/) B = (/) )
4	1, 3	eqtrd 2656	. . . 4 |- ( A = (/) -> U_ k e. A B = (/) )
5	4	adantl 482	. . 3 |- ( ( ph /\ A = (/) ) -> U_ k e. A B = (/) )
6		caragenfiiuncl.o	. . . . 5 |- ( ph -> O e. OutMeas )
7		caragenfiiuncl.s	. . . . 5 |- S = (CaraGen ` O )
8	6, 7	caragen0 40720	. . . 4 |- ( ph -> (/) e. S )
9	8	adantr 481	. . 3 |- ( ( ph /\ A = (/) ) -> (/) e. S )
10	5, 9	eqeltrd 2701	. 2 |- ( ( ph /\ A = (/) ) -> U_ k e. A B e. S )
11		simpl 473	. . 3 |- ( ( ph /\ -. A = (/) ) -> ph )
12		neqne 2802	. . . 4 |- ( -. A = (/) -> A =/= (/) )
13	12	adantl 482	. . 3 |- ( ( ph /\ -. A = (/) ) -> A =/= (/) )
14		caragenfiiuncl.kph	. . . . 5 |- F/ k ph
15		nfv 1843	. . . . 5 |- F/ k A =/= (/)
16	14, 15	nfan 1828	. . . 4 |- F/ k ( ph /\ A =/= (/) )
17		caragenfiiuncl.b	. . . . 5 |- ( ( ph /\ k e. A ) -> B e. S )
18	17	adantlr 751	. . . 4 |- ( ( ( ph /\ A =/= (/) ) /\ k e. A ) -> B e. S )
19	6	3ad2ant1 1082	. . . . . 6 |- ( ( ph /\ x e. S /\ y e. S ) -> O e. OutMeas )
20		simp2 1062	. . . . . 6 |- ( ( ph /\ x e. S /\ y e. S ) -> x e. S )
21		simp3 1063	. . . . . 6 |- ( ( ph /\ x e. S /\ y e. S ) -> y e. S )
22	19, 7, 20, 21	caragenuncl 40727	. . . . 5 |- ( ( ph /\ x e. S /\ y e. S ) -> ( x u. y ) e. S )
23	22	3adant1r 1319	. . . 4 |- ( ( ( ph /\ A =/= (/) ) /\ x e. S /\ y e. S ) -> ( x u. y ) e. S )
24		caragenfiiuncl.a	. . . . 5 |- ( ph -> A e. Fin )
25	24	adantr 481	. . . 4 |- ( ( ph /\ A =/= (/) ) -> A e. Fin )
26		simpr 477	. . . 4 |- ( ( ph /\ A =/= (/) ) -> A =/= (/) )
27	16, 18, 23, 25, 26	fiiuncl 39234	. . 3 |- ( ( ph /\ A =/= (/) ) -> U_ k e. A B e. S )
28	11, 13, 27	syl2anc 693	. 2 |- ( ( ph /\ -. A = (/) ) -> U_ k e. A B e. S )
29	10, 28	pm2.61dan 832	1 |- ( ph -> U_ k e. A B e. S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   =/= wne 2794   u. cun 3572   (/)c0 3915   U_ciun 4520   ` cfv 5888   Fincfn 7955  OutMeascome 40703  CaraGenccaragen 40705

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-xadd 11947  df-icc 12182  df-ome 40704  df-caragen 40706

This theorem is referenced by:  carageniuncllem1  40735  carageniuncllem2  40736