Theorem f1ovscpbl 13613

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

Step	Hyp	Ref	Expression
1		f1ocpbl.f	. . . . 5 |- ( ph -> F : V -1-1-onto-> X )
2		f1of1 5636	. . . . 5 |- ( F : V -1-1-onto-> X -> F : V -1-1-> X )
3	1, 2	syl 14	. . . 4 |- ( ph -> F : V -1-1-> X )
4	3	adantr 276	. . 3 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> F : V -1-1-> X )
5		simpr2 1035	. . 3 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> B e. V )
6		simpr3 1036	. . 3 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> C e. V )
7		f1fveq 5971	. . 3 |- ( ( F : V -1-1-> X /\ ( B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
8	4, 5, 6, 7	syl12anc 1276	. 2 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
9		oveq2 6086	. . 3 |- ( B = C -> ( A .+ B ) = ( A .+ C ) )
10	9	fveq2d 5697	. 2 |- ( B = C -> ( F ` ( A .+ B ) ) = ( F ` ( A .+ C ) ) )
11	8, 10	biimtrdi 163	1 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) -> ( F ` ( A .+ B ) ) = ( F ` ( A .+ C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   -1-1->wf1 5372   -1-1-onto->wf1o 5374   ` cfv 5375  (class class class)co 6078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-f1o 5382  df-fv 5383  df-ov 6081

This theorem is referenced by: (None)