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Theorem f1ovscpbl 13753
Description: An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypothesis
Ref Expression
f1ocpbl.f  |-  ( ph  ->  F : V -1-1-onto-> X )
Assertion
Ref Expression
f1ovscpbl  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V ) )  -> 
( ( F `  B )  =  ( F `  C )  ->  ( F `  ( A  .+  B ) )  =  ( F `
 ( A  .+  C ) ) ) )

Proof of Theorem f1ovscpbl
StepHypRef Expression
1 f1ocpbl.f . . . . 5  |-  ( ph  ->  F : V -1-1-onto-> X )
2 f1of1 5675 . . . . 5  |-  ( F : V -1-1-onto-> X  ->  F : V -1-1-> X )
31, 2syl 16 . . . 4  |-  ( ph  ->  F : V -1-1-> X
)
43adantr 453 . . 3  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V ) )  ->  F : V -1-1-> X )
5 simpr2 965 . . 3  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V ) )  ->  B  e.  V )
6 simpr3 966 . . 3  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V ) )  ->  C  e.  V )
7 f1fveq 6010 . . 3  |-  ( ( F : V -1-1-> X  /\  ( B  e.  V  /\  C  e.  V
) )  ->  (
( F `  B
)  =  ( F `
 C )  <->  B  =  C ) )
84, 5, 6, 7syl12anc 1183 . 2  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V ) )  -> 
( ( F `  B )  =  ( F `  C )  <-> 
B  =  C ) )
9 oveq2 6091 . . 3  |-  ( B  =  C  ->  ( A  .+  B )  =  ( A  .+  C
) )
109fveq2d 5734 . 2  |-  ( B  =  C  ->  ( F `  ( A  .+  B ) )  =  ( F `  ( A  .+  C ) ) )
118, 10syl6bi 221 1  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V ) )  -> 
( ( F `  B )  =  ( F `  C )  ->  ( F `  ( A  .+  B ) )  =  ( F `
 ( A  .+  C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   -1-1->wf1 5453   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083
This theorem is referenced by:  xpsvsca  13806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-f1o 5463  df-fv 5464  df-ov 6086
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