Theorem fun11uni 5188

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 108	. . . . 5 |- ( ( Fun f /\ Fun `' f ) -> Fun f )
2	1	anim1i 338	. . . 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 2493	. . 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 5186	. . 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 109	. . . . 5 |- ( ( Fun f /\ Fun `' f ) -> Fun `' f )
7	6	anim1i 338	. . . 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 2493	. . 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 5187	. . 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 304	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 103   \/ wo 697   A.wral 2414   C_ wss 3066   U.cuni 3731   `'ccnv 4533   Fun wfun 5112

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-fun 5120

This theorem is referenced by:  fun11iun  5381  ennnfonelemf1  11920