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Theorem tz7.48-2 7099
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 3518 . . 3  |-  On  C_  On
2 onelon 4898 . . . . . . . . 9  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
32ancoms 453 . . . . . . . 8  |-  ( ( y  e.  x  /\  x  e.  On )  ->  y  e.  On )
4 tz7.48.1 . . . . . . . . . . 11  |-  F  Fn  On
5 fndm 5673 . . . . . . . . . . 11  |-  ( F  Fn  On  ->  dom  F  =  On )
64, 5ax-mp 5 . . . . . . . . . 10  |-  dom  F  =  On
76eleq2i 2540 . . . . . . . . 9  |-  ( y  e.  dom  F  <->  y  e.  On )
8 fnfun 5671 . . . . . . . . . . . . 13  |-  ( F  Fn  On  ->  Fun  F )
94, 8ax-mp 5 . . . . . . . . . . . 12  |-  Fun  F
10 funfvima 6128 . . . . . . . . . . . 12  |-  ( ( Fun  F  /\  y  e.  dom  F )  -> 
( y  e.  x  ->  ( F `  y
)  e.  ( F
" x ) ) )
119, 10mpan 670 . . . . . . . . . . 11  |-  ( y  e.  dom  F  -> 
( y  e.  x  ->  ( F `  y
)  e.  ( F
" x ) ) )
1211impcom 430 . . . . . . . . . 10  |-  ( ( y  e.  x  /\  y  e.  dom  F )  ->  ( F `  y )  e.  ( F " x ) )
13 eleq1a 2545 . . . . . . . . . . 11  |-  ( ( F `  y )  e.  ( F "
x )  ->  (
( F `  x
)  =  ( F `
 y )  -> 
( F `  x
)  e.  ( F
" x ) ) )
14 eldifn 3622 . . . . . . . . . . 11  |-  ( ( F `  x )  e.  ( A  \ 
( F " x
) )  ->  -.  ( F `  x )  e.  ( F "
x ) )
1513, 14nsyli 141 . . . . . . . . . 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 476 . . . . . . . 8  |-  ( ( y  e.  x  /\  y  e.  On )  ->  ( ( F `  x )  e.  ( A  \  ( F
" x ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
183, 17syldan 470 . . . . . . 7  |-  ( ( y  e.  x  /\  x  e.  On )  ->  ( ( F `  x )  e.  ( A  \  ( F
" x ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
1918expimpd 603 . . . . . 6  |-  ( y  e.  x  ->  (
( x  e.  On  /\  ( F `  x
)  e.  ( A 
\  ( F "
x ) ) )  ->  -.  ( F `  x )  =  ( F `  y ) ) )
2019com12 31 . . . . 5  |-  ( ( x  e.  On  /\  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  -> 
( y  e.  x  ->  -.  ( F `  x )  =  ( F `  y ) ) )
2120ralrimiv 2871 . . . 4  |-  ( ( x  e.  On  /\  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  ->  A. y  e.  x  -.  ( F `  x
)  =  ( F `
 y ) )
2221ralimiaa 2851 . . 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 7098 . . 3  |-  ( ( On  C_  On  /\  A. x  e.  On  A. y  e.  x  -.  ( F `  x )  =  ( F `  y ) )  ->  Fun  `' ( F  |`  On ) )
241, 22, 23sylancr 663 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' ( F  |`  On ) )
25 fnrel 5672 . . . . . 6  |-  ( F  Fn  On  ->  Rel  F )
264, 25ax-mp 5 . . . . 5  |-  Rel  F
276eqimssi 3553 . . . . 5  |-  dom  F  C_  On
28 relssres 5304 . . . . 5  |-  ( ( Rel  F  /\  dom  F 
C_  On )  -> 
( F  |`  On )  =  F )
2926, 27, 28mp2an 672 . . . 4  |-  ( F  |`  On )  =  F
3029cnveqi 5170 . . 3  |-  `' ( F  |`  On )  =  `' F
3130funeqi 5601 . 2  |-  ( Fun  `' ( F  |`  On )  <->  Fun  `' F )
3224, 31sylib 196 1  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' F
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809    \ cdif 3468    C_ wss 3471   Oncon0 4873   `'ccnv 4993   dom cdm 4994    |` cres 4996   "cima 4997   Rel wrel 4999   Fun wfun 5575    Fn wfn 5576   ` cfv 5581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fv 5589
This theorem is referenced by:  tz7.48-3  7101
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