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Theorem elovimad 6336
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  |-  ( ph  ->  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 6299 . 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 5040 . . . 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  |-  ( ph  ->  Fun  F )
7 elovimad.4 . . . . 5  |-  ( ph  ->  ( C  X.  D
)  C_  dom  F )
87, 5sseldd 3500 . . . 4  |-  ( ph  -> 
<. A ,  B >.  e. 
dom  F )
9 funfvima 6148 . . . 4  |-  ( ( Fun  F  /\  <. A ,  B >.  e.  dom  F )  ->  ( <. A ,  B >.  e.  ( C  X.  D )  ->  ( F `  <. A ,  B >. )  e.  ( F "
( C  X.  D
) ) ) )
106, 8, 9syl2anc 661 . . 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 2549 1  |-  ( ph  ->  ( A F B )  e.  ( F
" ( C  X.  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819    C_ wss 3471   <.cop 4038    X. cxp 5006   dom cdm 5008   "cima 5011   Fun wfun 5588   ` cfv 5594  (class class class)co 6296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-ov 6299
This theorem is referenced by:  ltgov  24108  xrofsup  27734
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