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Theorem dff3im 5656
Description: Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.)
Assertion
Ref Expression
dff3im  |-  ( F : A --> B  -> 
( F  C_  ( A  X.  B )  /\  A. x  e.  A  E! y  x F y ) )
Distinct variable groups:    x, y, A   
x, B, y    x, F, y

Proof of Theorem dff3im
StepHypRef Expression
1 fssxp 5378 . 2  |-  ( F : A --> B  ->  F  C_  ( A  X.  B ) )
2 ffun 5363 . . . . . . . 8  |-  ( F : A --> B  ->  Fun  F )
32adantr 276 . . . . . . 7  |-  ( ( F : A --> B  /\  x  e.  A )  ->  Fun  F )
4 fdm 5366 . . . . . . . . 9  |-  ( F : A --> B  ->  dom  F  =  A )
54eleq2d 2247 . . . . . . . 8  |-  ( F : A --> B  -> 
( x  e.  dom  F  <-> 
x  e.  A ) )
65biimpar 297 . . . . . . 7  |-  ( ( F : A --> B  /\  x  e.  A )  ->  x  e.  dom  F
)
7 funfvop 5623 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  <. x ,  ( F `
 x ) >.  e.  F )
83, 6, 7syl2anc 411 . . . . . 6  |-  ( ( F : A --> B  /\  x  e.  A )  -> 
<. x ,  ( F `
 x ) >.  e.  F )
9 df-br 4001 . . . . . 6  |-  ( x F ( F `  x )  <->  <. x ,  ( F `  x
) >.  e.  F )
108, 9sylibr 134 . . . . 5  |-  ( ( F : A --> B  /\  x  e.  A )  ->  x F ( F `
 x ) )
11 funfvex 5527 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  _V )
12 breq2 4004 . . . . . . . 8  |-  ( y  =  ( F `  x )  ->  (
x F y  <->  x F
( F `  x
) ) )
1312spcegv 2825 . . . . . . 7  |-  ( ( F `  x )  e.  _V  ->  (
x F ( F `
 x )  ->  E. y  x F
y ) )
1411, 13syl 14 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( x F ( F `  x )  ->  E. y  x F y ) )
153, 6, 14syl2anc 411 . . . . 5  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( x F ( F `  x )  ->  E. y  x F y ) )
1610, 15mpd 13 . . . 4  |-  ( ( F : A --> B  /\  x  e.  A )  ->  E. y  x F y )
17 funmo 5226 . . . . . 6  |-  ( Fun 
F  ->  E* y  x F y )
182, 17syl 14 . . . . 5  |-  ( F : A --> B  ->  E* y  x F
y )
1918adantr 276 . . . 4  |-  ( ( F : A --> B  /\  x  e.  A )  ->  E* y  x F y )
20 eu5 2073 . . . 4  |-  ( E! y  x F y  <-> 
( E. y  x F y  /\  E* y  x F y ) )
2116, 19, 20sylanbrc 417 . . 3  |-  ( ( F : A --> B  /\  x  e.  A )  ->  E! y  x F y )
2221ralrimiva 2550 . 2  |-  ( F : A --> B  ->  A. x  e.  A  E! y  x F
y )
231, 22jca 306 1  |-  ( F : A --> B  -> 
( F  C_  ( A  X.  B )  /\  A. x  e.  A  E! y  x F y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   E.wex 1492   E!weu 2026   E*wmo 2027    e. wcel 2148   A.wral 2455   _Vcvv 2737    C_ wss 3129   <.cop 3594   class class class wbr 4000    X. cxp 4620   dom cdm 4622   Fun wfun 5205   -->wf 5207   ` cfv 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219
This theorem is referenced by:  dff4im  5657
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