Theorem elovmpt2wrd 30405

Description: Implications for the value of an operation defined by the maps-to notation with a class abstration of words as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Hypothesis

Ref	Expression
elovmpt2wrd.o	|- O = ( v e. _V , y e. _V |-> { z e. Word v | ph } )

Assertion

Ref	Expression
elovmpt2wrd	|- ( Z e. ( V O Y ) -> ( V e. _V /\ Y e. _V /\ Z e. Word V ) )

Distinct variable groups:   v, V, y, z   v, Y, y, z   z, Z

Allowed substitution hints:   ph( y, z, v)   O( y, z, v)   Z( y, v)

Proof of Theorem elovmpt2wrd

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elovmpt2wrd.o	. . . 4 |- O = ( v e. _V , y e. _V |-> { z e. Word v | ph } )
2		csbwrdg 12376	. . . . . . . 8 |- ( v e. _V -> [_ v / x ]_Word x = Word v )
3	2	eqcomd 2462	. . . . . . 7 |- ( v e. _V -> Word v = [_ v / x ]_Word x )
4	3	adantr 465	. . . . . 6 |- ( ( v e. _V /\ y e. _V ) -> Word v = [_ v / x ]_Word x )
5		rabeq 3072	. . . . . 6 |- (Word v = [_ v / x ]_Word x -> { z e. Word v | ph } = { z e. [_ v / x ]_Word x | ph } )
6	4, 5	syl 16	. . . . 5 |- ( ( v e. _V /\ y e. _V ) -> { z e. Word v | ph } = { z e. [_ v / x ]_Word x | ph } )
7	6	mpt2eq3ia 6261	. . . 4 |- ( v e. _V , y e. _V |-> { z e. Word v | ph } ) = ( v e. _V , y e. _V |-> { z e. [_ v / x ]_Word x | ph } )
8	1, 7	eqtri 2483	. . 3 |- O = ( v e. _V , y e. _V |-> { z e. [_ v / x ]_Word x | ph } )
9		csbwrdg 12376	. . . . 5 |- ( V e. _V -> [_ V / x ]_Word x = Word V )
10		wrdexg 12363	. . . . 5 |- ( V e. _V -> Word V e. _V )
11	9, 10	eqeltrd 2542	. . . 4 |- ( V e. _V -> [_ V / x ]_Word x e. _V )
12	11	adantr 465	. . 3 |- ( ( V e. _V /\ Y e. _V ) -> [_ V / x ]_Word x e. _V )
13	8, 12	elovmpt2rab1 30308	. 2 |- ( Z e. ( V O Y ) -> ( V e. _V /\ Y e. _V /\ Z e. [_ V / x ]_Word x ) )
14	9	eleq2d 2524	. . . . 5 |- ( V e. _V -> ( Z e. [_ V / x ]_Word x <-> Z e. Word V ) )
15	14	adantr 465	. . . 4 |- ( ( V e. _V /\ Y e. _V ) -> ( Z e. [_ V / x ]_Word x <-> Z e. Word V ) )
16		id 22	. . . . 5 |- ( ( V e. _V /\ Y e. _V /\ Z e. Word V ) -> ( V e. _V /\ Y e. _V /\ Z e. Word V ) )
17	16	3expia 1190	. . . 4 |- ( ( V e. _V /\ Y e. _V ) -> ( Z e. Word V -> ( V e. _V /\ Y e. _V /\ Z e. Word V ) ) )
18	15, 17	sylbid 215	. . 3 |- ( ( V e. _V /\ Y e. _V ) -> ( Z e. [_ V / x ]_Word x -> ( V e. _V /\ Y e. _V /\ Z e. Word V ) ) )
19	18	3impia 1185	. 2 |- ( ( V e. _V /\ Y e. _V /\ Z e. [_ V / x ]_Word x ) -> ( V e. _V /\ Y e. _V /\ Z e. Word V ) )
20	13, 19	syl 16	1 |- ( Z e. ( V O Y ) -> ( V e. _V /\ Y e. _V /\ Z e. Word V ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   {crab 2803   _Vcvv 3078   [_csb 3396  (class class class)co 6201   |-> cmpt2 6203  Word cword 12340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-map 7327  df-pm 7328  df-neg 9710  df-z 10759  df-uz 10974  df-fz 11556  df-fzo 11667  df-word 12348

This theorem is referenced by:  wwlkprop  30468