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Theorem tz7.49c 6412
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
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 6411 . 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 959 . . . 4  |-  ( ( A. y  e.  x  ( A  \  ( F " y ) )  =/=  (/)  /\  ( F
" x )  =  A  /\  Fun  `' ( F  |`  x ) )  ->  ( ( F " x )  =  A  /\  Fun  `' ( F  |`  x ) ) )
5 onss 4540 . . . . . . . . 9  |-  ( x  e.  On  ->  x  C_  On )
6 fnssres 5281 . . . . . . . . 9  |-  ( ( F  Fn  On  /\  x  C_  On )  -> 
( F  |`  x
)  Fn  x )
71, 5, 6sylancr 647 . . . . . . . 8  |-  ( x  e.  On  ->  ( F  |`  x )  Fn  x )
8 df-ima 4668 . . . . . . . . . 10  |-  ( F
" x )  =  ran  (  F  |`  x )
98eqeq1i 2263 . . . . . . . . 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 554 . . . . . 6  |-  ( ( ( x  e.  On  /\  ( F " x
)  =  A )  /\  Fun  `' ( F  |`  x )
)  ->  ( (
( F  |`  x
)  Fn  x  /\  ran  (  F  |`  x
)  =  A )  /\  Fun  `' ( F  |`  x )
) )
13 dff1o2 5401 . . . . . . 7  |-  ( ( F  |`  x ) : x -1-1-onto-> A  <->  ( ( F  |`  x )  Fn  x  /\  Fun  `' ( F  |`  x )  /\  ran  (  F  |`  x )  =  A ) )
14 3anan32 951 . . . . . . 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 604 . . . 4  |-  ( x  e.  On  ->  (
( ( F "
x )  =  A  /\  Fun  `' ( F  |`  x )
)  ->  ( F  |`  x ) : x -1-1-onto-> A ) )
184, 17syl5 30 . . 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 2621 . 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 17 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   A.wral 2516   E.wrex 2517    \ cdif 3110    C_ wss 3113   (/)c0 3416   Oncon0 4350   `'ccnv 4646   ran crn 4648    |` cres 4649   "cima 4650   Fun wfun 4653    Fn wfn 4654   -1-1-onto->wf1o 4658   ` cfv 4659
This theorem is referenced by:  dfac8alem  7610  dnnumch1  26494
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675
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