Theorem elfvmptrab1 5985

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 3728	. . 3 |- ( Y e. ( F ` X ) -> ( F ` X ) =/= (/) )
2		ndmfv 5903	. . . 4 |- ( -. X e. dom F -> ( F ` X ) = (/) )
3	2	necon1ai 2670	. . 3 |- ( ( F ` X ) =/= (/) -> X e. dom F )
4		elfvmptrab1.f	. . . . . . . 8 |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
5	4	dmmptss 5338	. . . . . . 7 |- dom F C_ V
6	5	sseli 3414	. . . . . 6 |- ( X e. dom F -> X e. V )
7		elfvmptrab1.v	. . . . . . 7 |- ( X e. V -> [_ X / m ]_ M e. _V )
8		rabexg 4549	. . . . . . 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 2612	. . . . . . 7 |- F/_ x X
11		nfsbc1v 3275	. . . . . . . 8 |- F/ x [. X / x ]. ph
12		nfcv 2612	. . . . . . . . 9 |- F/_ x M
13	10, 12	nfcsb 3367	. . . . . . . 8 |- F/_ x [_ X / m ]_ M
14	11, 13	nfrab 2958	. . . . . . 7 |- F/_ x { y e. [_ X / m ]_ M | [. X / x ]. ph }
15		csbeq1 3352	. . . . . . . 8 |- ( x = X -> [_ x / m ]_ M = [_ X / m ]_ M )
16		sbceq1a 3266	. . . . . . . 8 |- ( x = X -> ( ph <-> [. X / x ]. ph ) )
17	15, 16	rabeqbidv 3026	. . . . . . 7 |- ( x = X -> { y e. [_ x / m ]_ M | ph } = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
18	10, 14, 17, 4	fvmptf 5981	. . . . . 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 673	. . . . 5 |- ( X e. dom F -> ( F ` X ) = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
20	19	eleq2d 2534	. . . 4 |- ( X e. dom F -> ( Y e. ( F ` X ) <-> Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } ) )
21		elrabi 3181	. . . . . 6 |- ( Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } -> Y e. [_ X / m ]_ M )
22	6, 21	anim12i 576	. . . . 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 441	. . . 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 223	. . 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 48	1 |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   =/= wne 2641   {crab 2760   _Vcvv 3031   [.wsbc 3255   [_csb 3349   (/)c0 3722   |-> cmpt 4454   dom cdm 4839   ` cfv 5589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597

This theorem is referenced by:  elfvmptrab  5986