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Theorem tz7.49c 7014
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 7013 . 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 987 . . . 4  |-  ( ( A. y  e.  x  ( A  \  ( F " y ) )  =/=  (/)  /\  ( F
" x )  =  A  /\  Fun  `' ( F  |`  x ) )  ->  ( ( F " x )  =  A  /\  Fun  `' ( F  |`  x ) ) )
5 onss 6515 . . . . . . . . 9  |-  ( x  e.  On  ->  x  C_  On )
6 fnssres 5635 . . . . . . . . 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 4964 . . . . . . . . . 10  |-  ( F
" x )  =  ran  ( F  |`  x )
98eqeq1i 2461 . . . . . . . . 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 5757 . . . . . . 7  |-  ( ( F  |`  x ) : x -1-1-onto-> A  <->  ( ( F  |`  x )  Fn  x  /\  Fun  `' ( F  |`  x )  /\  ran  ( F  |`  x )  =  A ) )
14 3anan32 977 . . . . . . 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 2927 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800    \ cdif 3436    C_ wss 3439   (/)c0 3748   Oncon0 4830   `'ccnv 4950   ran crn 4952    |` cres 4953   "cima 4954   Fun wfun 5523    Fn wfn 5524   -1-1-onto->wf1o 5528   ` cfv 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537
This theorem is referenced by:  dfac8alem  8314  dnnumch1  29568
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