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Theorem fvsn2a 25115
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
fvsn2a  |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  A )  =  C )

Proof of Theorem fvsn2a
StepHypRef Expression
1 fvsn2.1 . . 3  |-  A  e.  E
21elexi 2797 . 2  |-  A  e. 
_V
3 fvsn2.3 . . 3  |-  C  e.  G
43elexi 2797 . 2  |-  C  e. 
_V
52, 4fvpr1 5722 1  |-  ( A  =/=  B  ->  ( { <. A ,  C >. ,  <. B ,  D >. } `  A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   {cpr 3641   <.cop 3643   ` cfv 5255
This theorem is referenced by:  repcpwti  25161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263
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