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Theorem tz7.49c 6732
Description: Corollary of Proposition 7.49 of [TakeutiZaring] p. 51. (Contributed by NM, 10-Feb-1997.) (Revised by Mario Carneiro, 19-Jan-2013.)
Hypothesis
Ref Expression
tz7.49c.1  |-  F  Fn  On
Assertion
Ref Expression
tz7.49c  |-  ( ( A  e.  B  /\  A. x  e.  On  (
( A  \  ( F " x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
Distinct variable groups:    x, A    x, F
Allowed substitution hint:    B( x)

Proof of Theorem tz7.49c
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 tz7.49c.1 . . 3  |-  F  Fn  On
2 biid 229 . . 3  |-  ( A. x  e.  On  (
( A  \  ( F " x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) )  <->  A. x  e.  On  ( ( A  \ 
( F " x
) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F " x ) ) ) )
31, 2tz7.49 6731 . 2  |-  ( ( A  e.  B  /\  A. x  e.  On  (
( A  \  ( F " x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )  ->  E. x  e.  On  ( A. y  e.  x  ( A  \  ( F " y
) )  =/=  (/)  /\  ( F " x )  =  A  /\  Fun  `' ( F  |`  x ) ) )
4 3simpc 957 . . . 4  |-  ( ( A. y  e.  x  ( A  \  ( F " y ) )  =/=  (/)  /\  ( F
" x )  =  A  /\  Fun  `' ( F  |`  x ) )  ->  ( ( F " x )  =  A  /\  Fun  `' ( F  |`  x ) ) )
5 onss 4800 . . . . . . . . 9  |-  ( x  e.  On  ->  x  C_  On )
6 fnssres 5587 . . . . . . . . 9  |-  ( ( F  Fn  On  /\  x  C_  On )  -> 
( F  |`  x
)  Fn  x )
71, 5, 6sylancr 646 . . . . . . . 8  |-  ( x  e.  On  ->  ( F  |`  x )  Fn  x )
8 df-ima 4920 . . . . . . . . . 10  |-  ( F
" x )  =  ran  ( F  |`  x )
98eqeq1i 2449 . . . . . . . . 9  |-  ( ( F " x )  =  A  <->  ran  ( F  |`  x )  =  A )
109biimpi 188 . . . . . . . 8  |-  ( ( F " x )  =  A  ->  ran  ( F  |`  x )  =  A )
117, 10anim12i 551 . . . . . . 7  |-  ( ( x  e.  On  /\  ( F " x )  =  A )  -> 
( ( F  |`  x )  Fn  x  /\  ran  ( F  |`  x )  =  A ) )
1211anim1i 553 . . . . . 6  |-  ( ( ( x  e.  On  /\  ( F " x
)  =  A )  /\  Fun  `' ( F  |`  x )
)  ->  ( (
( F  |`  x
)  Fn  x  /\  ran  ( F  |`  x
)  =  A )  /\  Fun  `' ( F  |`  x )
) )
13 dff1o2 5708 . . . . . . 7  |-  ( ( F  |`  x ) : x -1-1-onto-> A  <->  ( ( F  |`  x )  Fn  x  /\  Fun  `' ( F  |`  x )  /\  ran  ( F  |`  x )  =  A ) )
14 3anan32 949 . . . . . . 7  |-  ( ( ( F  |`  x
)  Fn  x  /\  Fun  `' ( F  |`  x )  /\  ran  ( F  |`  x )  =  A )  <->  ( (
( F  |`  x
)  Fn  x  /\  ran  ( F  |`  x
)  =  A )  /\  Fun  `' ( F  |`  x )
) )
1513, 14bitri 242 . . . . . 6  |-  ( ( F  |`  x ) : x -1-1-onto-> A  <->  ( ( ( F  |`  x )  Fn  x  /\  ran  ( F  |`  x )  =  A )  /\  Fun  `' ( F  |`  x
) ) )
1612, 15sylibr 205 . . . . 5  |-  ( ( ( x  e.  On  /\  ( F " x
)  =  A )  /\  Fun  `' ( F  |`  x )
)  ->  ( F  |`  x ) : x -1-1-onto-> A )
1716expl 603 . . . 4  |-  ( x  e.  On  ->  (
( ( F "
x )  =  A  /\  Fun  `' ( F  |`  x )
)  ->  ( F  |`  x ) : x -1-1-onto-> A ) )
184, 17syl5 31 . . 3  |-  ( x  e.  On  ->  (
( A. y  e.  x  ( A  \ 
( F " y
) )  =/=  (/)  /\  ( F " x )  =  A  /\  Fun  `' ( F  |`  x ) )  ->  ( F  |`  x ) : x -1-1-onto-> A ) )
1918reximia 2817 . 2  |-  ( E. x  e.  On  ( A. y  e.  x  ( A  \  ( F " y ) )  =/=  (/)  /\  ( F
" x )  =  A  /\  Fun  `' ( F  |`  x ) )  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
203, 19syl 16 1  |-  ( ( A  e.  B  /\  A. x  e.  On  (
( A  \  ( F " x ) )  =/=  (/)  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )  ->  E. x  e.  On  ( F  |`  x ) : x -1-1-onto-> A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711   E.wrex 2712    \ cdif 3303    C_ wss 3306   (/)c0 3613   Oncon0 4610   `'ccnv 4906   ran crn 4908    |` cres 4909   "cima 4910   Fun wfun 5477    Fn wfn 5478   -1-1-onto->wf1o 5482   ` cfv 5483
This theorem is referenced by:  dfac8alem  7941  dnnumch1  27157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491
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