Theorem elovmpt2wrd 12491

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. Note that ph may depend on z as well as on v and y. (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 12478	. . . . . . . 8 |- ( v e. _V -> [_ v / x ]_Word x = Word v )
3	2	eqcomd 2390	. . . . . . 7 |- ( v e. _V -> Word v = [_ v / x ]_Word x )
4	3	adantr 463	. . . . . 6 |- ( ( v e. _V /\ y e. _V ) -> Word v = [_ v / x ]_Word x )
5		rabeq 3028	. . . . . 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 2411	. . 3 |- O = ( v e. _V , y e. _V |-> { z e. [_ v / x ]_Word x | ph } )
9		csbwrdg 12478	. . . . 5 |- ( V e. _V -> [_ V / x ]_Word x = Word V )
10		wrdexg 12464	. . . . 5 |- ( V e. _V -> Word V e. _V )
11	9, 10	eqeltrd 2470	. . . 4 |- ( V e. _V -> [_ V / x ]_Word x e. _V )
12	11	adantr 463	. . 3 |- ( ( V e. _V /\ Y e. _V ) -> [_ V / x ]_Word x e. _V )
13	8, 12	elovmpt2rab1 6421	. 2 |- ( Z e. ( V O Y ) -> ( V e. _V /\ Y e. _V /\ Z e. [_ V / x ]_Word x ) )
14	9	eleq2d 2452	. . . . 5 |- ( V e. _V -> ( Z e. [_ V / x ]_Word x <-> Z e. Word V ) )
15	14	adantr 463	. . . 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 1196	. . . 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 1191	. 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 367   /\ w3a 971   = wceq 1399   e. wcel 1826   {crab 2736   _Vcvv 3034   [_csb 3348  (class class class)co 6196   |-> cmpt2 6198  Word cword 12438

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-map 7340  df-pm 7341  df-neg 9721  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-word 12446

This theorem is referenced by:  wwlkprop  24806