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Theorem elovimad 25956
Description: Elementhood of the image set of an operation value (Contributed by Thierry Arnoux, 13-Mar-2017.)
Hypotheses
Ref Expression
elovimad.1  |-  ( ph  ->  A  e.  C )
elovimad.2  |-  ( ph  ->  B  e.  D )
elovimad.3  |-  Fun  F
elovimad.4  |-  ( ph  ->  ( C  X.  D
)  C_  dom  F )
Assertion
Ref Expression
elovimad  |-  ( ph  ->  ( A F B )  e.  ( F
" ( C  X.  D ) ) )

Proof of Theorem elovimad
StepHypRef Expression
1 df-ov 6094 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 elovimad.1 . . . 4  |-  ( ph  ->  A  e.  C )
3 elovimad.2 . . . 4  |-  ( ph  ->  B  e.  D )
4 opelxpi 4871 . . . 4  |-  ( ( A  e.  C  /\  B  e.  D )  -> 
<. A ,  B >.  e.  ( C  X.  D
) )
52, 3, 4syl2anc 661 . . 3  |-  ( ph  -> 
<. A ,  B >.  e.  ( C  X.  D
) )
6 elovimad.3 . . . 4  |-  Fun  F
7 elovimad.4 . . . . 5  |-  ( ph  ->  ( C  X.  D
)  C_  dom  F )
87, 5sseldd 3357 . . . 4  |-  ( ph  -> 
<. A ,  B >.  e. 
dom  F )
9 funfvima 5952 . . . 4  |-  ( ( Fun  F  /\  <. A ,  B >.  e.  dom  F )  ->  ( <. A ,  B >.  e.  ( C  X.  D )  ->  ( F `  <. A ,  B >. )  e.  ( F "
( C  X.  D
) ) ) )
106, 8, 9sylancr 663 . . 3  |-  ( ph  ->  ( <. A ,  B >.  e.  ( C  X.  D )  ->  ( F `  <. A ,  B >. )  e.  ( F " ( C  X.  D ) ) ) )
115, 10mpd 15 . 2  |-  ( ph  ->  ( F `  <. A ,  B >. )  e.  ( F " ( C  X.  D ) ) )
121, 11syl5eqel 2527 1  |-  ( ph  ->  ( A F B )  e.  ( F
" ( C  X.  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1756    C_ wss 3328   <.cop 3883    X. cxp 4838   dom cdm 4840   "cima 4843   Fun wfun 5412   ` cfv 5418  (class class class)co 6091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-fv 5426  df-ov 6094
This theorem is referenced by:  xrofsup  26055
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