Theorem elovmpt2wrd 12557

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 12544	. . . . . . . 8 |- ( v e. _V -> [_ v / x ]_Word x = Word v )
3	2	eqcomd 2449	. . . . . . 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 3087	. . . . . 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 6343	. . . 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 2470	. . 3 |- O = ( v e. _V , y e. _V |-> { z e. [_ v / x ]_Word x | ph } )
9		csbwrdg 12544	. . . . 5 |- ( V e. _V -> [_ V / x ]_Word x = Word V )
10		wrdexg 12531	. . . . 5 |- ( V e. _V -> Word V e. _V )
11	9, 10	eqeltrd 2529	. . . 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 6503	. 2 |- ( Z e. ( V O Y ) -> ( V e. _V /\ Y e. _V /\ Z e. [_ V / x ]_Word x ) )
14	9	eleq2d 2511	. . . . 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 1197	. . . 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 1192	. 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 972   = wceq 1381   e. wcel 1802   {crab 2795   _Vcvv 3093   [_csb 3417  (class class class)co 6277   |-> cmpt2 6279  Word cword 12508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-map 7420  df-pm 7421  df-neg 9808  df-z 10866  df-uz 11086  df-fz 11677  df-fzo 11799  df-word 12516

This theorem is referenced by:  wwlkprop  24550