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Theorem f1ovscpbl 13428
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 5471 . . . . 5  |-  ( F : V -1-1-onto-> X  ->  F : V -1-1-> X )
31, 2syl 15 . . . 4  |-  ( ph  ->  F : V -1-1-> X
)
43adantr 451 . . 3  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V ) )  ->  F : V -1-1-> X )
5 simpr2 962 . . 3  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V ) )  ->  B  e.  V )
6 simpr3 963 . . 3  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V ) )  ->  C  e.  V )
7 f1fveq 5786 . . 3  |-  ( ( F : V -1-1-> X  /\  ( B  e.  V  /\  C  e.  V
) )  ->  (
( F `  B
)  =  ( F `
 C )  <->  B  =  C ) )
84, 5, 6, 7syl12anc 1180 . 2  |-  ( (
ph  /\  ( A  e.  K  /\  B  e.  V  /\  C  e.  V ) )  -> 
( ( F `  B )  =  ( F `  C )  <-> 
B  =  C ) )
9 oveq2 5866 . . 3  |-  ( B  =  C  ->  ( A  .+  B )  =  ( A  .+  C
) )
109fveq2d 5529 . 2  |-  ( B  =  C  ->  ( F `  ( A  .+  B ) )  =  ( F `  ( A  .+  C ) ) )
118, 10syl6bi 219 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858
This theorem is referenced by:  xpsvsca  13481
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-f1o 5262  df-fv 5263  df-ov 5861
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