Theorem elfvmptrab 5448

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

Hypotheses

Ref	Expression
elfvmptrab.f	|- F = ( x e. V |-> { y e. M | ph } )
elfvmptrab.v	|- ( X e. V -> M e. _V )

Assertion

Ref	Expression
elfvmptrab	|- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. M ) )

Distinct variable groups:   x, M, y   x, V   x, X, y   y, Y

Allowed substitution hints:   ph( x, y)   F( x, y)   V( y)   Y( x)

Proof of Theorem elfvmptrab

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvmptrab.f	. . . 4 |- F = ( x e. V |-> { y e. M | ph } )
2		csbconstg 2967	. . . . . . 7 |- ( x e. V -> [_ x / m ]_ M = M )
3	2	eqcomd 2105	. . . . . 6 |- ( x e. V -> M = [_ x / m ]_ M )
4		rabeq 2633	. . . . . 6 |- ( M = [_ x / m ]_ M -> { y e. M | ph } = { y e. [_ x / m ]_ M | ph } )
5	3, 4	syl 14	. . . . 5 |- ( x e. V -> { y e. M | ph } = { y e. [_ x / m ]_ M | ph } )
6	5	mpteq2ia 3954	. . . 4 |- ( x e. V |-> { y e. M | ph } ) = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
7	1, 6	eqtri 2120	. . 3 |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
8		csbconstg 2967	. . . 4 |- ( X e. V -> [_ X / m ]_ M = M )
9		elfvmptrab.v	. . . 4 |- ( X e. V -> M e. _V )
10	8, 9	eqeltrd 2176	. . 3 |- ( X e. V -> [_ X / m ]_ M e. _V )
11	7, 10	elfvmptrab1 5447	. 2 |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )
12	8	eleq2d 2169	. . . 4 |- ( X e. V -> ( Y e. [_ X / m ]_ M <-> Y e. M ) )
13	12	biimpd 143	. . 3 |- ( X e. V -> ( Y e. [_ X / m ]_ M -> Y e. M ) )
14	13	imdistani 437	. 2 |- ( ( X e. V /\ Y e. [_ X / m ]_ M ) -> ( X e. V /\ Y e. M ) )
15	11, 14	syl 14	1 |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. M ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   e. wcel 1448   {crab 2379   _Vcvv 2641   [_csb 2955   |-> cmpt 3929   ` cfv 5059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fv 5067

This theorem is referenced by: (None)