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Theorem fun11uni 5334
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
StepHypRef Expression
1 simpl 443 . . . . 5  |-  ( ( Fun  f  /\  Fun  `' f )  ->  Fun  f )
21anim1i 551 . . . 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 ) ) )
32ralimi 2631 . . 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 5332 . . 3  |-  ( A. f  e.  A  ( Fun  f  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f ) )  ->  Fun  U. A )
53, 4syl 15 . 2  |-  ( A. f  e.  A  (
( Fun  f  /\  Fun  `' f )  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f ) )  ->  Fun  U. A )
6 simpr 447 . . . . 5  |-  ( ( Fun  f  /\  Fun  `' f )  ->  Fun  `' f )
76anim1i 551 . . . 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 ) ) )
87ralimi 2631 . . 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 5333 . . 3  |-  ( A. f  e.  A  ( Fun  `' f  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f ) )  ->  Fun  `' U. A )
108, 9syl 15 . 2  |-  ( A. f  e.  A  (
( Fun  f  /\  Fun  `' f )  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f ) )  ->  Fun  `' U. A
)
115, 10jca 518 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 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   A.wral 2556    C_ wss 3165   U.cuni 3843   `'ccnv 4704   Fun wfun 5265
This theorem is referenced by:  fun11iun  5509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273
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