Theorem f1ovscpbl 13475

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 5591	. . . . 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 1031	. . 3 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> B e. V )
6		simpr3 1032	. . 3 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> C e. V )
7		f1fveq 5923	. . 3 |- ( ( F : V -1-1-> X /\ ( B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
8	4, 5, 6, 7	syl12anc 1272	. 2 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
9		oveq2 6036	. . 3 |- ( B = C -> ( A .+ B ) = ( A .+ C ) )
10	9	fveq2d 5652	. 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 1005   = wceq 1398   e. wcel 2202   -1-1->wf1 5330   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-f1o 5340  df-fv 5341  df-ov 6031

This theorem is referenced by: (None)