Theorem elfvmptrab 5751

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 3142	. . . . . . 7 |- ( x e. V -> [_ x / m ]_ M = M )
3	2	eqcomd 2237	. . . . . 6 |- ( x e. V -> M = [_ x / m ]_ M )
4		rabeq 2795	. . . . . 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 4180	. . . 4 |- ( x e. V |-> { y e. M | ph } ) = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
7	1, 6	eqtri 2252	. . 3 |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
8		csbconstg 3142	. . . 4 |- ( X e. V -> [_ X / m ]_ M = M )
9		elfvmptrab.v	. . . 4 |- ( X e. V -> M e. _V )
10	8, 9	eqeltrd 2308	. . 3 |- ( X e. V -> [_ X / m ]_ M e. _V )
11	7, 10	elfvmptrab1 5750	. 2 |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )
12	8	eleq2d 2301	. . . 4 |- ( X e. V -> ( Y e. [_ X / m ]_ M <-> Y e. M ) )
13	12	biimpd 144	. . 3 |- ( X e. V -> ( Y e. [_ X / m ]_ M -> Y e. M ) )
14	13	imdistani 445	. 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 104   = wceq 1398   e. wcel 2202   {crab 2515   _Vcvv 2803   [_csb 3128   |-> cmpt 4155   ` cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fv 5341

This theorem is referenced by: (None)