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Theorem fun11uni 5519
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 444 . . . . 5  |-  ( ( Fun  f  /\  Fun  `' f )  ->  Fun  f )
21anim1i 552 . . . 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 2781 . . 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 5517 . . 3  |-  ( A. f  e.  A  ( Fun  f  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f ) )  ->  Fun  U. A )
53, 4syl 16 . 2  |-  ( A. f  e.  A  (
( Fun  f  /\  Fun  `' f )  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f ) )  ->  Fun  U. A )
6 simpr 448 . . . . 5  |-  ( ( Fun  f  /\  Fun  `' f )  ->  Fun  `' f )
76anim1i 552 . . . 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 2781 . . 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 5518 . . 3  |-  ( A. f  e.  A  ( Fun  `' f  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f ) )  ->  Fun  `' U. A )
108, 9syl 16 . 2  |-  ( A. f  e.  A  (
( Fun  f  /\  Fun  `' f )  /\  A. g  e.  A  ( f  C_  g  \/  g  C_  f ) )  ->  Fun  `' U. A
)
115, 10jca 519 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 358    /\ wa 359   A.wral 2705    C_ wss 3320   U.cuni 4015   `'ccnv 4877   Fun wfun 5448
This theorem is referenced by:  fun11iun  5695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-fun 5456
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