Theorem f1ovscpbl 12895

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 5499	. . . . 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 1006	. . 3 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> B e. V )
6		simpr3 1007	. . 3 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> C e. V )
7		f1fveq 5815	. . 3 |- ( ( F : V -1-1-> X /\ ( B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
8	4, 5, 6, 7	syl12anc 1247	. 2 |- ( ( ph /\ ( A e. K /\ B e. V /\ C e. V ) ) -> ( ( F ` B ) = ( F ` C ) <-> B = C ) )
9		oveq2 5926	. . 3 |- ( B = C -> ( A .+ B ) = ( A .+ C ) )
10	9	fveq2d 5558	. 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 980   = wceq 1364   e. wcel 2164   -1-1->wf1 5251   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-f1o 5261  df-fv 5262  df-ov 5921

This theorem is referenced by: (None)