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Theorem tz7.48-2 6422
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
StepHypRef Expression
1 ssid 3172 . . 3  |-  On  C_  On
2 onelon 4389 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
32ancoms 441 . . . . . . . 8  |-  ( ( y  e.  x  /\  x  e.  On )  ->  y  e.  On )
4 tz7.48.1 . . . . . . . . . . 11  |-  F  Fn  On
5 fndm 5281 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  dom  F  =  On )
64, 5ax-mp 10 . . . . . . . . . 10  |-  dom  F  =  On
76eleq2i 2322 . . . . . . . . 9  |-  ( y  e.  dom  F  <->  y  e.  On )
8 fnfun 5279 . . . . . . . . . . . . 13  |-  ( F  Fn  On  ->  Fun  F )
94, 8ax-mp 10 . . . . . . . . . . . 12  |-  Fun  F
10 funfvima 5687 . . . . . . . . . . . 12  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( y  e.  x  ->  ( F `  y
)  e.  ( F
" x ) ) )
119, 10mpan 654 . . . . . . . . . . 11  |-  ( y  e.  dom  F  -> 
( y  e.  x  ->  ( F `  y
)  e.  ( F
" x ) ) )
1211impcom 421 . . . . . . . . . 10  |-  ( ( y  e.  x  /\  y  e.  dom  F )  ->  ( F `  y )  e.  ( F " x ) )
13 eleq1a 2327 . . . . . . . . . . 11  |-  ( ( F `  y )  e.  ( F "
x )  ->  (
( F `  x
)  =  ( F `
 y )  -> 
( F `  x
)  e.  ( F
" x ) ) )
14 eldifn 3274 . . . . . . . . . . 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 17 . . . . . . . . 9  |-  ( ( y  e.  x  /\  y  e.  dom  F )  ->  ( ( F `
 x )  e.  ( A  \  ( F " x ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
177, 16sylan2br 464 . . . . . . . 8  |-  ( ( y  e.  x  /\  y  e.  On )  ->  ( ( F `  x )  e.  ( A  \  ( F
" x ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
183, 17syldan 458 . . . . . . 7  |-  ( ( y  e.  x  /\  x  e.  On )  ->  ( ( F `  x )  e.  ( A  \  ( F
" x ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
1918expimpd 589 . . . . . 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 2600 . . . 4  |-  ( ( x  e.  On  /\  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  ->  A. y  e.  x  -.  ( F `  x
)  =  ( F `
 y ) )
2221ralimiaa 2592 . . 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 6421 . . 3  |-  ( ( On  C_  On  /\  A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y ) )  ->  Fun  `' ( F  |`  On ) )
241, 22, 23sylancr 647 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' ( F  |`  On ) )
25 fnrel 5280 . . . . . 6  |-  ( F  Fn  On  ->  Rel  F )
264, 25ax-mp 10 . . . . 5  |-  Rel  F
276eqimssi 3207 . . . . 5  |-  dom  F  C_  On
28 relssres 4980 . . . . 5  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
2926, 27, 28mp2an 656 . . . 4  |-  ( F  |`  On )  =  F
3029cnveqi 4844 . . 3  |-  `' ( F  |`  On )  =  `' F
3130funeqi 5214 . 2  |-  ( Fun  `' ( F  |`  On )  <->  Fun  `' F )
3224, 31sylib 190 1  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' F
)
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2518    \ cdif 3124    C_ wss 3127   Oncon0 4364   `'ccnv 4660   dom cdm 4661    |` cres 4663   "cima 4664   Rel wrel 4666   Fun wfun 4667    Fn wfn 4668   ` cfv 4673
This theorem is referenced by:  tz7.48-3  6424
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fv 4689
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