Theorem iunon 6449

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 4989	. . 3 |- ( A. x e. A B e. On -> U_ x e. A B = U. ran ( x e. A |-> B ) )
2	1	adantl 277	. 2 |- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B = U. ran ( x e. A |-> B ) )
3		mptexg 5878	. . . 4 |- ( A e. V -> ( x e. A |-> B ) e. _V )
4		rnexg 4997	. . . 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 2231	. . . . 5 |- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	fmpt 5797	. . . 4 |- ( A. x e. A B e. On <-> ( x e. A |-> B ) : A --> On )
8		frn 5491	. . . 4 |- ( ( x e. A |-> B ) : A --> On -> ran ( x e. A |-> B ) C_ On )
9	7, 8	sylbi 121	. . 3 |- ( A. x e. A B e. On -> ran ( x e. A |-> B ) C_ On )
10		ssonuni 4586	. . . 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 124	. . 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 289	. 2 |- ( ( A e. V /\ A. x e. A B e. On ) -> U. ran ( x e. A |-> B ) e. On )
13	2, 12	eqeltrd 2308	1 |- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B e. On )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   U.cuni 3893   U_ciun 3970   |-> cmpt 4150   Oncon0 4460   ran crn 4726   -->wf 5322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334

This theorem is referenced by:  rdgon  6551