Theorem pfxval 41383

Description: Value of a prefix. (Contributed by AV, 2-May-2020.)

Assertion

Ref	Expression
pfxval	|- ( ( S e. V /\ L e. NN0 ) -> ( S prefix L ) = ( S substr <. 0 , L >. ) )

Proof of Theorem pfxval

Dummy variables l s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pfx 41382	. . 3 |- prefix = ( s e. _V , l e. NN0 |-> ( s substr <. 0 , l >. ) )
2	1	a1i 11	. 2 |- ( ( S e. V /\ L e. NN0 ) -> prefix = ( s e. _V , l e. NN0 |-> ( s substr <. 0 , l >. ) ) )
3		simpl 473	. . . 4 |- ( ( s = S /\ l = L ) -> s = S )
4		opeq2 4403	. . . . 5 |- ( l = L -> <. 0 , l >. = <. 0 , L >. )
5	4	adantl 482	. . . 4 |- ( ( s = S /\ l = L ) -> <. 0 , l >. = <. 0 , L >. )
6	3, 5	oveq12d 6668	. . 3 |- ( ( s = S /\ l = L ) -> ( s substr <. 0 , l >. ) = ( S substr <. 0 , L >. ) )
7	6	adantl 482	. 2 |- ( ( ( S e. V /\ L e. NN0 ) /\ ( s = S /\ l = L ) ) -> ( s substr <. 0 , l >. ) = ( S substr <. 0 , L >. ) )
8		elex 3212	. . 3 |- ( S e. V -> S e. _V )
9	8	adantr 481	. 2 |- ( ( S e. V /\ L e. NN0 ) -> S e. _V )
10		simpr 477	. 2 |- ( ( S e. V /\ L e. NN0 ) -> L e. NN0 )
11		ovexd 6680	. 2 |- ( ( S e. V /\ L e. NN0 ) -> ( S substr <. 0 , L >. ) e. _V )
12	2, 7, 9, 10, 11	ovmpt2d 6788	1 |- ( ( S e. V /\ L e. NN0 ) -> ( S prefix L ) = ( S substr <. 0 , L >. ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   NN0cn0 11292   substr csubstr 13295   prefix cpfx 41381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pfx 41382

This theorem is referenced by:  pfx00  41384  pfx0  41385  pfxcl  41386  pfxmpt  41387  pfxid  41392  pfxn0  41394  pfxnd  41395  pfxfv  41399  pfx1  41411  pfxswrd  41413  swrdpfx  41414  pfxpfx  41415  pfxccatpfx1  41427  pfxccatpfx2  41428  splvalpfx  41435  cshword2  41437  pfxco  41438