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Theorem f1ocpbl 14463
Description: An injection is compatible with any operations on the base set. (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypothesis
Ref Expression
f1ocpbl.f  |-  ( ph  ->  F : V -1-1-onto-> X )
Assertion
Ref Expression
f1ocpbl  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( (
( F `  A
)  =  ( F `
 C )  /\  ( F `  B )  =  ( F `  D ) )  -> 
( F `  ( A  .+  B ) )  =  ( F `  ( C  .+  D ) ) ) )

Proof of Theorem f1ocpbl
StepHypRef Expression
1 f1ocpbl.f . . 3  |-  ( ph  ->  F : V -1-1-onto-> X )
21f1ocpbllem 14462 . 2  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( (
( F `  A
)  =  ( F `
 C )  /\  ( F `  B )  =  ( F `  D ) )  <->  ( A  =  C  /\  B  =  D ) ) )
3 oveq12 6100 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  .+  B
)  =  ( C 
.+  D ) )
43fveq2d 5695 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( F `  ( A  .+  B ) )  =  ( F `  ( C  .+  D ) ) )
52, 4syl6bi 228 1  |-  ( (
ph  /\  ( A  e.  V  /\  B  e.  V )  /\  ( C  e.  V  /\  D  e.  V )
)  ->  ( (
( F `  A
)  =  ( F `
 C )  /\  ( F `  B )  =  ( F `  D ) )  -> 
( F `  ( A  .+  B ) )  =  ( F `  ( C  .+  D ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-f1o 5425  df-fv 5426  df-ov 6094
This theorem is referenced by:  xpsadd  14514  xpsmul  14515  imasmndf1  15460  imasgrpf1  15672  imasgim  29455
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