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Theorem fun11uni 5318
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 2618 . . 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 5316 . . 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 2618 . . 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 5317 . . 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 2543    C_ wss 3152   U.cuni 3827   `'ccnv 4688   Fun wfun 5249
This theorem is referenced by:  fun11iun  5493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257
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