Theorem iunon 6555

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 5039	. . 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 5942	. . . 4 |- ( A e. V -> ( x e. A |-> B ) e. _V )
4		rnexg 5047	. . . 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 2238	. . . . 5 |- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	fmpt 5858	. . . 4 |- ( A. x e. A B e. On <-> ( x e. A |-> B ) : A --> On )
8		frn 5542	. . . 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 4635	. . . 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 2315	1 |- ( ( A e. V /\ A. x e. A B e. On ) -> U_ x e. A B e. On )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   U.cuni 3935   U_ciun 4012   |-> cmpt 4192   Oncon0 4508   ran crn 4775   -->wf 5373

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385

This theorem is used by:  rdgon  6657