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Theorem tz7.48-2 6691
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 3359 . . 3  |-  On  C_  On
2 onelon 4598 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
32ancoms 440 . . . . . . . 8  |-  ( ( y  e.  x  /\  x  e.  On )  ->  y  e.  On )
4 tz7.48.1 . . . . . . . . . . 11  |-  F  Fn  On
5 fndm 5536 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  dom  F  =  On )
64, 5ax-mp 8 . . . . . . . . . 10  |-  dom  F  =  On
76eleq2i 2499 . . . . . . . . 9  |-  ( y  e.  dom  F  <->  y  e.  On )
8 fnfun 5534 . . . . . . . . . . . . 13  |-  ( F  Fn  On  ->  Fun  F )
94, 8ax-mp 8 . . . . . . . . . . . 12  |-  Fun  F
10 funfvima 5965 . . . . . . . . . . . 12  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( y  e.  x  ->  ( F `  y
)  e.  ( F
" x ) ) )
119, 10mpan 652 . . . . . . . . . . 11  |-  ( y  e.  dom  F  -> 
( y  e.  x  ->  ( F `  y
)  e.  ( F
" x ) ) )
1211impcom 420 . . . . . . . . . 10  |-  ( ( y  e.  x  /\  y  e.  dom  F )  ->  ( F `  y )  e.  ( F " x ) )
13 eleq1a 2504 . . . . . . . . . . 11  |-  ( ( F `  y )  e.  ( F "
x )  ->  (
( F `  x
)  =  ( F `
 y )  -> 
( F `  x
)  e.  ( F
" x ) ) )
14 eldifn 3462 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ( A  \ 
( F " x
) )  ->  -.  ( F `  x )  e.  ( F "
x ) )
1513, 14nsyli 135 . . . . . . . . . 10  |-  ( ( F `  y )  e.  ( F "
x )  ->  (
( F `  x
)  e.  ( A 
\  ( F "
x ) )  ->  -.  ( F `  x
)  =  ( F `
 y ) ) )
1612, 15syl 16 . . . . . . . . 9  |-  ( ( y  e.  x  /\  y  e.  dom  F )  ->  ( ( F `
 x )  e.  ( A  \  ( F " x ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
177, 16sylan2br 463 . . . . . . . 8  |-  ( ( y  e.  x  /\  y  e.  On )  ->  ( ( F `  x )  e.  ( A  \  ( F
" x ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
183, 17syldan 457 . . . . . . 7  |-  ( ( y  e.  x  /\  x  e.  On )  ->  ( ( F `  x )  e.  ( A  \  ( F
" x ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
1918expimpd 587 . . . . . 6  |-  ( y  e.  x  ->  (
( x  e.  On  /\  ( F `  x
)  e.  ( A 
\  ( F "
x ) ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
2019com12 29 . . . . 5  |-  ( ( x  e.  On  /\  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  -> 
( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
2120ralrimiv 2780 . . . 4  |-  ( ( x  e.  On  /\  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  ->  A. y  e.  x  -.  ( F `  x
)  =  ( F `
 y ) )
2221ralimiaa 2772 . . 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 6690 . . 3  |-  ( ( On  C_  On  /\  A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y ) )  ->  Fun  `' ( F  |`  On ) )
241, 22, 23sylancr 645 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' ( F  |`  On ) )
25 fnrel 5535 . . . . . 6  |-  ( F  Fn  On  ->  Rel  F )
264, 25ax-mp 8 . . . . 5  |-  Rel  F
276eqimssi 3394 . . . . 5  |-  dom  F  C_  On
28 relssres 5175 . . . . 5  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
2926, 27, 28mp2an 654 . . . 4  |-  ( F  |`  On )  =  F
3029cnveqi 5039 . . 3  |-  `' ( F  |`  On )  =  `' F
3130funeqi 5466 . 2  |-  ( Fun  `' ( F  |`  On )  <->  Fun  `' F )
3224, 31sylib 189 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 359    = wceq 1652    e. wcel 1725   A.wral 2697    \ cdif 3309    C_ wss 3312   Oncon0 4573   `'ccnv 4869   dom cdm 4870    |` cres 4872   "cima 4873   Rel wrel 4875   Fun wfun 5440    Fn wfn 5441   ` cfv 5446
This theorem is referenced by:  tz7.48-3  6693
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fv 5454
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