Theorem f1ovscpbl 13345

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 5571	. . . . 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 1028	. . 3 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> B e. V )
6		simpr3 1029	. . 3 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> C e. V )
7		f1fveq 5896	. . 3 |- ( ( F : V -1-1-> X /\ ( B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
8	4, 5, 6, 7	syl12anc 1269	. 2 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
9		oveq2 6009	. . 3 |- ( B = C -> ( A .+ B ) = ( A .+ C ) )
10	9	fveq2d 5631	. 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 1002   = wceq 1395   e. wcel 2200   -1-1->wf1 5315   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6001

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-f1o 5325  df-fv 5326  df-ov 6004

This theorem is referenced by: (None)