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Theorem tz7.49c 7123
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 236 . . 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 7122 . 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 995 . . . 4  |-  ( ( A. y  e.  x  ( A  \  ( F " y ) )  =/=  (/)  /\  ( F
" x )  =  A  /\  Fun  `' ( F  |`  x ) )  ->  ( ( F " x )  =  A  /\  Fun  `' ( F  |`  x ) ) )
5 onss 6621 . . . . . . . . 9  |-  ( x  e.  On  ->  x  C_  On )
6 fnssres 5700 . . . . . . . . 9  |-  ( ( F  Fn  On  /\  x  C_  On )  -> 
( F  |`  x
)  Fn  x )
71, 5, 6sylancr 663 . . . . . . . 8  |-  ( x  e.  On  ->  ( F  |`  x )  Fn  x )
8 df-ima 5018 . . . . . . . . . 10  |-  ( F
" x )  =  ran  ( F  |`  x )
98eqeq1i 2474 . . . . . . . . 9  |-  ( ( F " x )  =  A  <->  ran  ( F  |`  x )  =  A )
109biimpi 194 . . . . . . . 8  |-  ( ( F " x )  =  A  ->  ran  ( F  |`  x )  =  A )
117, 10anim12i 566 . . . . . . 7  |-  ( ( x  e.  On  /\  ( F " x )  =  A )  -> 
( ( F  |`  x )  Fn  x  /\  ran  ( F  |`  x )  =  A ) )
1211anim1i 568 . . . . . 6  |-  ( ( ( x  e.  On  /\  ( F " x
)  =  A )  /\  Fun  `' ( F  |`  x )
)  ->  ( (
( F  |`  x
)  Fn  x  /\  ran  ( F  |`  x
)  =  A )  /\  Fun  `' ( F  |`  x )
) )
13 dff1o2 5827 . . . . . . 7  |-  ( ( F  |`  x ) : x -1-1-onto-> A  <->  ( ( F  |`  x )  Fn  x  /\  Fun  `' ( F  |`  x )  /\  ran  ( F  |`  x )  =  A ) )
14 3anan32 985 . . . . . . 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 249 . . . . . 6  |-  ( ( F  |`  x ) : x -1-1-onto-> A  <->  ( ( ( F  |`  x )  Fn  x  /\  ran  ( F  |`  x )  =  A )  /\  Fun  `' ( F  |`  x
) ) )
1612, 15sylibr 212 . . . . 5  |-  ( ( ( x  e.  On  /\  ( F " x
)  =  A )  /\  Fun  `' ( F  |`  x )
)  ->  ( F  |`  x ) : x -1-1-onto-> A )
1716expl 618 . . . 4  |-  ( x  e.  On  ->  (
( ( F "
x )  =  A  /\  Fun  `' ( F  |`  x )
)  ->  ( F  |`  x ) : x -1-1-onto-> A ) )
184, 17syl5 32 . . 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 2933 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    \ cdif 3478    C_ wss 3481   (/)c0 3790   Oncon0 4884   `'ccnv 5004   ran crn 5006    |` cres 5007   "cima 5008   Fun wfun 5588    Fn wfn 5589   -1-1-onto->wf1o 5593   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602
This theorem is referenced by:  dfac8alem  8422  dnnumch1  30918
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