Theorem iunon 5922

Description: The indexed union of a set of ordinal numbers B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003.) (Revised by Mario Carneiro, 5-Dec-2016.)

Assertion

Ref	Expression
iunon	|- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B e. On )

Distinct variable group:   x, A

Allowed substitution hints:   B( x)   V( x)

Proof of Theorem iunon

Step	Hyp	Ref	Expression
1		dfiun3g 4607	. . 3 |- ( A. x e. A B e. On -> U_ x e. A B = U. ran ( x e. A |-> B ) )
2	1	adantl 271	. 2 |- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B = U. ran ( x e. A |-> B ) )
3		mptexg 5407	. . . 4 |- ( A e. V -> ( x e. A |-> B ) e. _V )
4		rnexg 4615	. . . 4 |- ( ( x e. A |-> B ) e. _V -> ran ( x e. A |-> B ) e. _V )
5	3, 4	syl 14	. . 3 |- ( A e. V -> ran ( x e. A |-> B ) e. _V )
6		eqid 2081	. . . . 5 |- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	fmpt 5340	. . . 4 |- ( A. x e. A B e. On <-> ( x e. A |-> B ) : A --> On )
8		frn 5072	. . . 4 |- ( ( x e. A |-> B ) : A --> On -> ran ( x e. A |-> B ) C_ On )
9	7, 8	sylbi 119	. . 3 |- ( A. x e. A B e. On -> ran ( x e. A |-> B ) C_ On )
10		ssonuni 4232	. . . 4 |- ( ran ( x e. A |-> B ) e. _V -> ( ran ( x e. A |-> B ) C_ On -> U. ran ( x e. A |-> B ) e. On ) )
11	10	imp 122	. . 3 |- ( ( ran ( x e. A |-> B ) e. _V /\ ran ( x e. A |-> B ) C_ On ) -> U. ran ( x e. A |-> B ) e. On )
12	5, 9, 11	syl2an 283	. 2 |- ( ( A e. V /\ A. x e. A B e. On ) -> U. ran ( x e. A |-> B ) e. On )
13	2, 12	eqeltrd 2155	1 |- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B e. On )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   _Vcvv 2601   C_ wss 2973   U.cuni 3601   U_ciun 3678   |-> cmpt 3839   Oncon0 4118   ran crn 4364   -->wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930

This theorem is referenced by:  rdgon  5996