Theorem elfvmptrab1 6305

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

Hypotheses

Ref	Expression
elfvmptrab1.f	|- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
elfvmptrab1.v	|- ( X e. V -> [_ X / m ]_ M e. _V )

Assertion

Ref	Expression
elfvmptrab1	|- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )

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

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

Proof of Theorem elfvmptrab1

Step	Hyp	Ref	Expression
1		ne0i 3921	. . 3 |- ( Y e. ( F ` X ) -> ( F ` X ) =/= (/) )
2		ndmfv 6218	. . . 4 |- ( -. X e. dom F -> ( F ` X ) = (/) )
3	2	necon1ai 2821	. . 3 |- ( ( F ` X ) =/= (/) -> X e. dom F )
4		elfvmptrab1.f	. . . . . . . 8 |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
5	4	dmmptss 5631	. . . . . . 7 |- dom F C_ V
6	5	sseli 3599	. . . . . 6 |- ( X e. dom F -> X e. V )
7		elfvmptrab1.v	. . . . . . 7 |- ( X e. V -> [_ X / m ]_ M e. _V )
8		rabexg 4812	. . . . . . 7 |- ( [_ X / m ]_ M e. _V -> { y e. [_ X / m ]_ M | [. X / x ]. ph } e. _V )
9	6, 7, 8	3syl 18	. . . . . 6 |- ( X e. dom F -> { y e. [_ X / m ]_ M | [. X / x ]. ph } e. _V )
10		nfcv 2764	. . . . . . 7 |- F/_ x X
11		nfsbc1v 3455	. . . . . . . 8 |- F/ x [. X / x ]. ph
12		nfcv 2764	. . . . . . . . 9 |- F/_ x M
13	10, 12	nfcsb 3551	. . . . . . . 8 |- F/_ x [_ X / m ]_ M
14	11, 13	nfrab 3123	. . . . . . 7 |- F/_ x { y e. [_ X / m ]_ M | [. X / x ]. ph }
15		csbeq1 3536	. . . . . . . 8 |- ( x = X -> [_ x / m ]_ M = [_ X / m ]_ M )
16		sbceq1a 3446	. . . . . . . 8 |- ( x = X -> ( ph <-> [. X / x ]. ph ) )
17	15, 16	rabeqbidv 3195	. . . . . . 7 |- ( x = X -> { y e. [_ x / m ]_ M | ph } = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
18	10, 14, 17, 4	fvmptf 6301	. . . . . 6 |- ( ( X e. V /\ { y e. [_ X / m ]_ M | [. X / x ]. ph } e. _V ) -> ( F ` X ) = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
19	6, 9, 18	syl2anc 693	. . . . 5 |- ( X e. dom F -> ( F ` X ) = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
20	19	eleq2d 2687	. . . 4 |- ( X e. dom F -> ( Y e. ( F ` X ) <-> Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } ) )
21		elrabi 3359	. . . . . 6 |- ( Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } -> Y e. [_ X / m ]_ M )
22	6, 21	anim12i 590	. . . . 5 |- ( ( X e. dom F /\ Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )
23	22	ex 450	. . . 4 |- ( X e. dom F -> ( Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } -> ( X e. V /\ Y e. [_ X / m ]_ M ) ) )
24	20, 23	sylbid 230	. . 3 |- ( X e. dom F -> ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) ) )
25	1, 3, 24	3syl 18	. 2 |- ( Y e. ( F ` X ) -> ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) ) )
26	25	pm2.43i 52	1 |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   [.wsbc 3435   [_csb 3533   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  elfvmptrab  6306