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Theorem tz7.48-3 6342
Description: Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
Hypothesis
Ref Expression
tz7.48.1  |-  F  Fn  On
Assertion
Ref Expression
tz7.48-3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  -.  A  e.  _V )
Distinct variable groups:    x, F    x, A

Proof of Theorem tz7.48-3
StepHypRef Expression
1 onprc 4467 . . . 4  |-  -.  On  e.  _V
2 tz7.48.1 . . . . . 6  |-  F  Fn  On
3 fndm 5200 . . . . . 6  |-  ( F  Fn  On  ->  dom  F  =  On )
42, 3ax-mp 10 . . . . 5  |-  dom  F  =  On
54eleq1i 2316 . . . 4  |-  ( dom 
F  e.  _V  <->  On  e.  _V )
61, 5mtbir 292 . . 3  |-  -.  dom  F  e.  _V
72tz7.48-2 6340 . . . 4  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' F
)
8 funrnex 5599 . . . . . 6  |-  ( dom  `'  F  e.  _V  ->  ( Fun  `' F  ->  ran  `'  F  e. 
_V ) )
98com12 29 . . . . 5  |-  ( Fun  `' F  ->  ( dom  `'  F  e.  _V  ->  ran  `'  F  e. 
_V ) )
10 df-rn 4599 . . . . . 6  |-  ran  F  =  dom  `'  F
1110eleq1i 2316 . . . . 5  |-  ( ran 
F  e.  _V  <->  dom  `'  F  e.  _V )
12 dfdm4 4779 . . . . . 6  |-  dom  F  =  ran  `'  F
1312eleq1i 2316 . . . . 5  |-  ( dom 
F  e.  _V  <->  ran  `'  F  e.  _V )
149, 11, 133imtr4g 263 . . . 4  |-  ( Fun  `' F  ->  ( ran 
F  e.  _V  ->  dom 
F  e.  _V )
)
157, 14syl 17 . . 3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( ran  F  e.  _V  ->  dom  F  e. 
_V ) )
166, 15mtoi 171 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  -.  ran  F  e. 
_V )
172tz7.48-1 6341 . . 3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ran  F  C_  A
)
18 ssexg 4057 . . . 4  |-  ( ( ran  F  C_  A  /\  A  e.  _V )  ->  ran  F  e.  _V )
1918ex 425 . . 3  |-  ( ran 
F  C_  A  ->  ( A  e.  _V  ->  ran 
F  e.  _V )
)
2017, 19syl 17 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( A  e. 
_V  ->  ran  F  e.  _V ) )
2116, 20mtod 170 1  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  -.  A  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    = wceq 1619    e. wcel 1621   A.wral 2509   _Vcvv 2727    \ cdif 3075    C_ wss 3078   Oncon0 4285   `'ccnv 4579   dom cdm 4580   ran crn 4581   "cima 4583   Fun wfun 4586    Fn wfn 4587   ` cfv 4592
This theorem is referenced by:  tz7.49  6343
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608
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