Theorem iunon 6252

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 4861	. . 3 |- ( A. x e. A B e. On -> U_ x e. A B = U. ran ( x e. A |-> B ) )
2	1	adantl 275	. 2 |- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B = U. ran ( x e. A |-> B ) )
3		mptexg 5710	. . . 4 |- ( A e. V -> ( x e. A |-> B ) e. _V )
4		rnexg 4869	. . . 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 2165	. . . . 5 |- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	fmpt 5635	. . . 4 |- ( A. x e. A B e. On <-> ( x e. A |-> B ) : A --> On )
8		frn 5346	. . . 4 |- ( ( x e. A |-> B ) : A --> On -> ran ( x e. A |-> B ) C_ On )
9	7, 8	sylbi 120	. . 3 |- ( A. x e. A B e. On -> ran ( x e. A |-> B ) C_ On )
10		ssonuni 4465	. . . 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 123	. . 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 287	. 2 |- ( ( A e. V /\ A. x e. A B e. On ) -> U. ran ( x e. A |-> B ) e. On )
13	2, 12	eqeltrd 2243	1 |- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B e. On )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   _Vcvv 2726   C_ wss 3116   U.cuni 3789   U_ciun 3866   |-> cmpt 4043   Oncon0 4341   ran crn 4605   -->wf 5184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by:  rdgon  6354