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Theorem tz7.49c 6458
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 227 . . 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 6457 . 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 954 . . . 4  |-  ( ( A. y  e.  x  ( A  \  ( F " y ) )  =/=  (/)  /\  ( F
" x )  =  A  /\  Fun  `' ( F  |`  x ) )  ->  ( ( F " x )  =  A  /\  Fun  `' ( F  |`  x ) ) )
5 onss 4582 . . . . . . . . 9  |-  ( x  e.  On  ->  x  C_  On )
6 fnssres 5357 . . . . . . . . 9  |-  ( ( F  Fn  On  /\  x  C_  On )  -> 
( F  |`  x
)  Fn  x )
71, 5, 6sylancr 644 . . . . . . . 8  |-  ( x  e.  On  ->  ( F  |`  x )  Fn  x )
8 df-ima 4702 . . . . . . . . . 10  |-  ( F
" x )  =  ran  ( F  |`  x )
98eqeq1i 2290 . . . . . . . . 9  |-  ( ( F " x )  =  A  <->  ran  ( F  |`  x )  =  A )
109biimpi 186 . . . . . . . 8  |-  ( ( F " x )  =  A  ->  ran  ( F  |`  x )  =  A )
117, 10anim12i 549 . . . . . . 7  |-  ( ( x  e.  On  /\  ( F " x )  =  A )  -> 
( ( F  |`  x )  Fn  x  /\  ran  ( F  |`  x )  =  A ) )
1211anim1i 551 . . . . . 6  |-  ( ( ( x  e.  On  /\  ( F " x
)  =  A )  /\  Fun  `' ( F  |`  x )
)  ->  ( (
( F  |`  x
)  Fn  x  /\  ran  ( F  |`  x
)  =  A )  /\  Fun  `' ( F  |`  x )
) )
13 dff1o2 5477 . . . . . . 7  |-  ( ( F  |`  x ) : x -1-1-onto-> A  <->  ( ( F  |`  x )  Fn  x  /\  Fun  `' ( F  |`  x )  /\  ran  ( F  |`  x )  =  A ) )
14 3anan32 946 . . . . . . 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 240 . . . . . 6  |-  ( ( F  |`  x ) : x -1-1-onto-> A  <->  ( ( ( F  |`  x )  Fn  x  /\  ran  ( F  |`  x )  =  A )  /\  Fun  `' ( F  |`  x
) ) )
1612, 15sylibr 203 . . . . 5  |-  ( ( ( x  e.  On  /\  ( F " x
)  =  A )  /\  Fun  `' ( F  |`  x )
)  ->  ( F  |`  x ) : x -1-1-onto-> A )
1716expl 601 . . . 4  |-  ( x  e.  On  ->  (
( ( F "
x )  =  A  /\  Fun  `' ( F  |`  x )
)  ->  ( F  |`  x ) : x -1-1-onto-> A ) )
184, 17syl5 28 . . 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 2648 . 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 15 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    \ cdif 3149    C_ wss 3152   (/)c0 3455   Oncon0 4392   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   -1-1-onto->wf1o 5254   ` cfv 5255
This theorem is referenced by:  dfac8alem  7656  dnnumch1  27141
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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