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Theorem tz7.48-2 6456
Description: Proposition 7.48(2) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.) (Revised by David Abernethy, 5-May-2013.)
Hypothesis
Ref Expression
tz7.48.1  |-  F  Fn  On
Assertion
Ref Expression
tz7.48-2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' F
)
Distinct variable group:    x, F
Allowed substitution hint:    A( x)

Proof of Theorem tz7.48-2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssid 3199 . . 3  |-  On  C_  On
2 onelon 4419 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
32ancoms 439 . . . . . . . 8  |-  ( ( y  e.  x  /\  x  e.  On )  ->  y  e.  On )
4 tz7.48.1 . . . . . . . . . . 11  |-  F  Fn  On
5 fndm 5345 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  dom  F  =  On )
64, 5ax-mp 8 . . . . . . . . . 10  |-  dom  F  =  On
76eleq2i 2349 . . . . . . . . 9  |-  ( y  e.  dom  F  <->  y  e.  On )
8 fnfun 5343 . . . . . . . . . . . . 13  |-  ( F  Fn  On  ->  Fun  F )
94, 8ax-mp 8 . . . . . . . . . . . 12  |-  Fun  F
10 funfvima 5755 . . . . . . . . . . . 12  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( y  e.  x  ->  ( F `  y
)  e.  ( F
" x ) ) )
119, 10mpan 651 . . . . . . . . . . 11  |-  ( y  e.  dom  F  -> 
( y  e.  x  ->  ( F `  y
)  e.  ( F
" x ) ) )
1211impcom 419 . . . . . . . . . 10  |-  ( ( y  e.  x  /\  y  e.  dom  F )  ->  ( F `  y )  e.  ( F " x ) )
13 eleq1a 2354 . . . . . . . . . . 11  |-  ( ( F `  y )  e.  ( F "
x )  ->  (
( F `  x
)  =  ( F `
 y )  -> 
( F `  x
)  e.  ( F
" x ) ) )
14 eldifn 3301 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ( A  \ 
( F " x
) )  ->  -.  ( F `  x )  e.  ( F "
x ) )
1513, 14nsyli 133 . . . . . . . . . 10  |-  ( ( F `  y )  e.  ( F "
x )  ->  (
( F `  x
)  e.  ( A 
\  ( F "
x ) )  ->  -.  ( F `  x
)  =  ( F `
 y ) ) )
1612, 15syl 15 . . . . . . . . 9  |-  ( ( y  e.  x  /\  y  e.  dom  F )  ->  ( ( F `
 x )  e.  ( A  \  ( F " x ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
177, 16sylan2br 462 . . . . . . . 8  |-  ( ( y  e.  x  /\  y  e.  On )  ->  ( ( F `  x )  e.  ( A  \  ( F
" x ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
183, 17syldan 456 . . . . . . 7  |-  ( ( y  e.  x  /\  x  e.  On )  ->  ( ( F `  x )  e.  ( A  \  ( F
" x ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
1918expimpd 586 . . . . . 6  |-  ( y  e.  x  ->  (
( x  e.  On  /\  ( F `  x
)  e.  ( A 
\  ( F "
x ) ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
2019com12 27 . . . . 5  |-  ( ( x  e.  On  /\  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  -> 
( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
2120ralrimiv 2627 . . . 4  |-  ( ( x  e.  On  /\  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  ->  A. y  e.  x  -.  ( F `  x
)  =  ( F `
 y ) )
2221ralimiaa 2619 . . 3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y ) )
234tz7.48lem 6455 . . 3  |-  ( ( On  C_  On  /\  A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y ) )  ->  Fun  `' ( F  |`  On ) )
241, 22, 23sylancr 644 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' ( F  |`  On ) )
25 fnrel 5344 . . . . . 6  |-  ( F  Fn  On  ->  Rel  F )
264, 25ax-mp 8 . . . . 5  |-  Rel  F
276eqimssi 3234 . . . . 5  |-  dom  F  C_  On
28 relssres 4994 . . . . 5  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
2926, 27, 28mp2an 653 . . . 4  |-  ( F  |`  On )  =  F
3029cnveqi 4858 . . 3  |-  `' ( F  |`  On )  =  `' F
3130funeqi 5277 . 2  |-  ( Fun  `' ( F  |`  On )  <->  Fun  `' F )
3224, 31sylib 188 1  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' F
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   A.wral 2545    \ cdif 3151    C_ wss 3154   Oncon0 4394   `'ccnv 4690   dom cdm 4691    |` cres 4693   "cima 4694   Rel wrel 4696   Fun wfun 5251    Fn wfn 5252   ` cfv 5257
This theorem is referenced by:  tz7.48-3  6458
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fv 5265
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