Theorem elfvmptrab 5575

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 3054	. . . . . . 7 |- ( x e. V -> [_ x / m ]_ M = M )
3	2	eqcomd 2170	. . . . . 6 |- ( x e. V -> M = [_ x / m ]_ M )
4		rabeq 2713	. . . . . 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 4062	. . . 4 |- ( x e. V |-> { y e. M | ph } ) = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
7	1, 6	eqtri 2185	. . 3 |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
8		csbconstg 3054	. . . 4 |- ( X e. V -> [_ X / m ]_ M = M )
9		elfvmptrab.v	. . . 4 |- ( X e. V -> M e. _V )
10	8, 9	eqeltrd 2241	. . 3 |- ( X e. V -> [_ X / m ]_ M e. _V )
11	7, 10	elfvmptrab1 5574	. 2 |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )
12	8	eleq2d 2234	. . . 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 1342   e. wcel 2135   {crab 2446   _Vcvv 2721   [_csb 3040   |-> cmpt 4037   ` cfv 5182

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fv 5190

This theorem is referenced by: (None)