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Theorem fnovrn2 25153
Description: A function's value belongs to its range. A more general version of fnovrn 6011. To be used with partial operations. (Contributed by FL, 10-Mar-2008.)
Assertion
Ref Expression
fnovrn2  |-  ( ( Fun  F  /\  <. A ,  B >.  e.  dom  F )  ->  ( A F B )  e.  ran  F )

Proof of Theorem fnovrn2
StepHypRef Expression
1 df-ov 5877 . 2  |-  ( A F B )  =  ( F `  <. A ,  B >. )
2 fvelrn 5677 . 2  |-  ( ( Fun  F  /\  <. A ,  B >.  e.  dom  F )  ->  ( F `  <. A ,  B >. )  e.  ran  F
)
31, 2syl5eqel 2380 1  |-  ( ( Fun  F  /\  <. A ,  B >.  e.  dom  F )  ->  ( A F B )  e.  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   <.cop 3656   dom cdm 4705   ran crn 4706   Fun wfun 5265   ` cfv 5271  (class class class)co 5874
This theorem is referenced by:  cmpmorp  25882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5877
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