Theorem elfvmptrab 5509

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 3011	. . . . . . 7 |- ( x e. V -> [_ x / m ]_ M = M )
3	2	eqcomd 2143	. . . . . 6 |- ( x e. V -> M = [_ x / m ]_ M )
4		rabeq 2673	. . . . . 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 4009	. . . 4 |- ( x e. V |-> { y e. M | ph } ) = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
7	1, 6	eqtri 2158	. . 3 |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
8		csbconstg 3011	. . . 4 |- ( X e. V -> [_ X / m ]_ M = M )
9		elfvmptrab.v	. . . 4 |- ( X e. V -> M e. _V )
10	8, 9	eqeltrd 2214	. . 3 |- ( X e. V -> [_ X / m ]_ M e. _V )
11	7, 10	elfvmptrab1 5508	. 2 |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )
12	8	eleq2d 2207	. . . 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 441	. 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 1331   e. wcel 1480   {crab 2418   _Vcvv 2681   [_csb 2998   |-> cmpt 3984   ` cfv 5118

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fv 5126

This theorem is referenced by: (None)