Theorem iunon 6515

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 5014	. . 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 5911	. . . 4 |- ( A e. V -> ( x e. A |-> B ) e. _V )
4		rnexg 5022	. . . 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 2232	. . . . 5 |- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	fmpt 5827	. . . 4 |- ( A. x e. A B e. On <-> ( x e. A |-> B ) : A --> On )
8		frn 5517	. . . 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 4610	. . . 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 2309	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 1398   e. wcel 2203   A.wral 2520   _Vcvv 2813   C_ wss 3211   U.cuni 3914   U_ciun 3991   |-> cmpt 4171   Oncon0 4484   ran crn 4750   -->wf 5348

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360

This theorem is referenced by:  rdgon  6617