Theorem ccatvalfn 13365

Description: The concatenation of two words is a function over the half-open integer range having the sum of the lengths of the word as length. (Contributed by Alexander van der Vekens, 30-Mar-2018.)

Assertion

Ref	Expression
ccatvalfn	|- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) Fn ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) )

Proof of Theorem ccatvalfn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatfval 13358	. 2 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) = ( x e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) |-> if ( x e. ( 0..^ ( # ` A ) ) , ( A ` x ) , ( B ` ( x - ( # ` A ) ) ) ) ) )
2		fvex 6201	. . . . 5 |- ( A ` x ) e. _V
3		fvex 6201	. . . . 5 |- ( B ` ( x - ( # ` A ) ) ) e. _V
4	2, 3	ifex 4156	. . . 4 |- if ( x e. ( 0..^ ( # ` A ) ) , ( A ` x ) , ( B ` ( x - ( # ` A ) ) ) ) e. _V
5		eqid 2622	. . . 4 |- ( x e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) |-> if ( x e. ( 0..^ ( # ` A ) ) , ( A ` x ) , ( B ` ( x - ( # ` A ) ) ) ) ) = ( x e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) |-> if ( x e. ( 0..^ ( # ` A ) ) , ( A ` x ) , ( B ` ( x - ( # ` A ) ) ) ) )
6	4, 5	fnmpti 6022	. . 3 |- ( x e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) |-> if ( x e. ( 0..^ ( # ` A ) ) , ( A ` x ) , ( B ` ( x - ( # ` A ) ) ) ) ) Fn ( 0..^ ( ( # ` A ) + ( # ` B ) ) )
7		fneq1 5979	. . 3 |- ( ( A ++ B ) = ( x e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) |-> if ( x e. ( 0..^ ( # ` A ) ) , ( A ` x ) , ( B ` ( x - ( # ` A ) ) ) ) ) -> ( ( A ++ B ) Fn ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) <-> ( x e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) |-> if ( x e. ( 0..^ ( # ` A ) ) , ( A ` x ) , ( B ` ( x - ( # ` A ) ) ) ) ) Fn ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) ) )
8	6, 7	mpbiri 248	. 2 |- ( ( A ++ B ) = ( x e. ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) |-> if ( x e. ( 0..^ ( # ` A ) ) , ( A ` x ) , ( B ` ( x - ( # ` A ) ) ) ) ) -> ( A ++ B ) Fn ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) )
9	1, 8	syl 17	1 |- ( ( A e. Word V /\ B e. Word V ) -> ( A ++ B ) Fn ( 0..^ ( ( # ` A ) + ( # ` B ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   |-> cmpt 4729   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   - cmin 10266  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293

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-rep 4771  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-concat 13301

This theorem is referenced by:  ccatlid  13369  ccatrid  13370  ccatrn  13372  swrdccatin12  13491  pfxccat1  41410  pfxccatin12  41425