Theorem fun11uni 5288

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 2540	. . 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 5286	. . 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 2540	. . 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 5287	. . 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 708   A.wral 2455   C_ wss 3131   U.cuni 3811   `'ccnv 4627   Fun wfun 5212

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220

This theorem is referenced by:  fun11iun  5484  ennnfonelemf1  12422