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Theorem fvsn2b 25219
Description: Value of a function with a domain of two different values. (Contributed by FL, 26-Jun-2011.)
Hypotheses
Ref Expression
fvsn2.1  |-  A  e.  E
fvsn2.2  |-  B  e.  F
fvsn2.3  |-  C  e.  G
fvsn2.4  |-  D  e.  H
Assertion
Ref Expression
fvsn2b  |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  B )  =  D )

Proof of Theorem fvsn2b
StepHypRef Expression
1 fvsn2.2 . . 3  |-  B  e.  F
21elexi 2810 . 2  |-  B  e. 
_V
3 fvsn2.4 . . 3  |-  D  e.  H
43elexi 2810 . 2  |-  D  e. 
_V
52, 4fvpr2 5739 1  |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  B )  =  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   {cpr 3654   <.cop 3656   ` cfv 5271
This theorem is referenced by:  repcpwti  25264
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-13 1698  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-pow 4204  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-res 4717  df-iota 5235  df-fun 5273  df-fv 5279
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