Theorem elfvmptrab 5581

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 3059	. . . . . . 7 |- ( x e. V -> [_ x / m ]_ M = M )
3	2	eqcomd 2171	. . . . . 6 |- ( x e. V -> M = [_ x / m ]_ M )
4		rabeq 2718	. . . . . 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 4068	. . . 4 |- ( x e. V |-> { y e. M | ph } ) = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
7	1, 6	eqtri 2186	. . 3 |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
8		csbconstg 3059	. . . 4 |- ( X e. V -> [_ X / m ]_ M = M )
9		elfvmptrab.v	. . . 4 |- ( X e. V -> M e. _V )
10	8, 9	eqeltrd 2243	. . 3 |- ( X e. V -> [_ X / m ]_ M e. _V )
11	7, 10	elfvmptrab1 5580	. 2 |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )
12	8	eleq2d 2236	. . . 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 442	. 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 1343   e. wcel 2136   {crab 2448   _Vcvv 2726   [_csb 3045   |-> cmpt 4043   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fv 5196

This theorem is referenced by: (None)