Theorem fun11uni 5363

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 2571	. . 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 5361	. . 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 2571	. . 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 5362	. . 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 710   A.wral 2486   C_ wss 3174   U.cuni 3864   `'ccnv 4692   Fun wfun 5284

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-fun 5292

This theorem is referenced by:  fun11iun  5565  ennnfonelemf1  12904