Theorem fun11uni 5343

Description: The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004.)

Assertion

Ref	Expression
fun11uni	|- ( A. f e. A ( ( Fun f /\ Fun `' f ) /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> ( Fun U. A /\ Fun `' U. A ) )

Distinct variable group:   f, g, A

Proof of Theorem fun11uni

Step	Hyp	Ref	Expression
1		simpl 109	. . . . 5 |- ( ( Fun f /\ Fun `' f ) -> Fun f )
2	1	anim1i 340	. . . 4 |- ( ( ( Fun f /\ Fun `' f ) /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> ( Fun f /\ A. g e. A ( f C_ g \/ g C_ f ) ) )
3	2	ralimi 2568	. . 3 |- ( A. f e. A ( ( Fun f /\ Fun `' f ) /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> A. f e. A ( Fun f /\ A. g e. A ( f C_ g \/ g C_ f ) ) )
4		fununi 5341	. . 3 |- ( A. f e. A ( Fun f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> Fun U. A )
5	3, 4	syl 14	. 2 |- ( A. f e. A ( ( Fun f /\ Fun `' f ) /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> Fun U. A )
6		simpr 110	. . . . 5 |- ( ( Fun f /\ Fun `' f ) -> Fun `' f )
7	6	anim1i 340	. . . 4 |- ( ( ( Fun f /\ Fun `' f ) /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) )
8	7	ralimi 2568	. . 3 |- ( A. f e. A ( ( Fun f /\ Fun `' f ) /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) )
9		funcnvuni 5342	. . 3 |- ( A. f e. A ( Fun `' f /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> Fun `' U. A )
10	8, 9	syl 14	. 2 |- ( A. f e. A ( ( Fun f /\ Fun `' f ) /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> Fun `' U. A )
11	5, 10	jca 306	1 |- ( A. f e. A ( ( Fun f /\ Fun `' f ) /\ A. g e. A ( f C_ g \/ g C_ f ) ) -> ( Fun U. A /\ Fun `' U. A ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   A.wral 2483   C_ wss 3165   U.cuni 3849   `'ccnv 4673   Fun wfun 5264

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-fun 5272

This theorem is referenced by:  fun11iun  5542  ennnfonelemf1  12731