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Theorem tz7.48-2 6454
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 3197 . . 3  |-  On  C_  On
2 onelon 4417 . . . . . . . . 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 5343 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  dom  F  =  On )
64, 5ax-mp 8 . . . . . . . . . 10  |-  dom  F  =  On
76eleq2i 2347 . . . . . . . . 9  |-  ( y  e.  dom  F  <->  y  e.  On )
8 fnfun 5341 . . . . . . . . . . . . 13  |-  ( F  Fn  On  ->  Fun  F )
94, 8ax-mp 8 . . . . . . . . . . . 12  |-  Fun  F
10 funfvima 5753 . . . . . . . . . . . 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 2352 . . . . . . . . . . 11  |-  ( ( F `  y )  e.  ( F "
x )  ->  (
( F `  x
)  =  ( F `
 y )  -> 
( F `  x
)  e.  ( F
" x ) ) )
14 eldifn 3299 . . . . . . . . . . 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 2625 . . . 4  |-  ( ( x  e.  On  /\  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  ->  A. y  e.  x  -.  ( F `  x
)  =  ( F `
 y ) )
2221ralimiaa 2617 . . 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 6453 . . 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 5342 . . . . . 6  |-  ( F  Fn  On  ->  Rel  F )
264, 25ax-mp 8 . . . . 5  |-  Rel  F
276eqimssi 3232 . . . . 5  |-  dom  F  C_  On
28 relssres 4992 . . . . 5  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
2926, 27, 28mp2an 653 . . . 4  |-  ( F  |`  On )  =  F
3029cnveqi 4856 . . 3  |-  `' ( F  |`  On )  =  `' F
3130funeqi 5275 . 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 1623    e. wcel 1684   A.wral 2543    \ cdif 3149    C_ wss 3152   Oncon0 4392   `'ccnv 4688   dom cdm 4689    |` cres 4691   "cima 4692   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  tz7.48-3  6456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fv 5263
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