Theorem f1ocpbl 13632

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

Step	Hyp	Ref	Expression
1		f1ocpbl.f	. . 3 |- ( ph -> F : V -1-1-onto-> X )
2	1	f1ocpbllem 13631	. 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 6094	. . 3 |- ( ( A = C /\ B = D ) -> ( A .+ B ) = ( C .+ D ) )
4	3	fveq2d 5699	. 2 |- ( ( A = C /\ B = D ) -> ( F ` ( A .+ B ) ) = ( F ` ( C .+ D ) ) )
5	2, 4	biimtrdi 163	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 ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-f1o 5384  df-fv 5385  df-ov 6088

This theorem is used by:  imasmndf1  13761  imasgrpf1  13915  imasrngf1  14256  imasringf1  14370