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Theorem tz7.48-3 6453
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 4577 . . . 4  |-  -.  On  e.  _V
2 tz7.48.1 . . . . . 6  |-  F  Fn  On
3 fndm 5310 . . . . . 6  |-  ( F  Fn  On  ->  dom  F  =  On )
42, 3ax-mp 10 . . . . 5  |-  dom  F  =  On
54eleq1i 2349 . . . 4  |-  ( dom 
F  e.  _V  <->  On  e.  _V )
61, 5mtbir 292 . . 3  |-  -.  dom  F  e.  _V
72tz7.48-2 6451 . . . 4  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' F
)
8 funrnex 5710 . . . . . 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 4701 . . . . . 6  |-  ran  F  =  dom  `'  F
1110eleq1i 2349 . . . . 5  |-  ( ran 
F  e.  _V  <->  dom  `'  F  e.  _V )
12 dfdm4 4873 . . . . . 6  |-  dom  F  =  ran  `'  F
1312eleq1i 2349 . . . . 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 6452 . . 3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ran  F  C_  A
)
18 ssexg 4163 . . . 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 1625    e. wcel 1687   A.wral 2546   _Vcvv 2791    \ cdif 3152    C_ wss 3155   Oncon0 4393   `'ccnv 4689   dom cdm 4690   ran crn 4691   "cima 4693   Fun wfun 5217    Fn wfn 5218   ` cfv 5223
This theorem is referenced by:  tz7.49  6454
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pr 4215  ax-un 4513
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-rex 2552  df-reu 2553  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-iun 3910  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231
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