Theorem elfvmptrab1 5970

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 3791	. . 3 |- ( Y e. ( F ` X ) -> ( F ` X ) =/= (/) )
2		ndmfv 5890	. . . 4 |- ( -. X e. dom F -> ( F ` X ) = (/) )
3	2	necon1ai 2698	. . 3 |- ( ( F ` X ) =/= (/) -> X e. dom F )
4		elfvmptrab1.f	. . . . . . . 8 |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
5	4	dmmptss 5503	. . . . . . 7 |- dom F C_ V
6	5	sseli 3500	. . . . . 6 |- ( X e. dom F -> X e. V )
7		elfvmptrab1.v	. . . . . . 7 |- ( X e. V -> [_ X / m ]_ M e. _V )
8		rabexg 4597	. . . . . . 7 |- ( [_ X / m ]_ M e. _V -> { y e. [_ X / m ]_ M | [. X / x ]. ph } e. _V )
9	6, 7, 8	3syl 20	. . . . . 6 |- ( X e. dom F -> { y e. [_ X / m ]_ M | [. X / x ]. ph } e. _V )
10		nfcv 2629	. . . . . . 7 |- F/_ x X
11		nfsbc1v 3351	. . . . . . . 8 |- F/ x [. X / x ]. ph
12		nfcv 2629	. . . . . . . . 9 |- F/_ x M
13	10, 12	nfcsb 3453	. . . . . . . 8 |- F/_ x [_ X / m ]_ M
14	11, 13	nfrab 3043	. . . . . . 7 |- F/_ x { y e. [_ X / m ]_ M | [. X / x ]. ph }
15		csbeq1 3438	. . . . . . . 8 |- ( x = X -> [_ x / m ]_ M = [_ X / m ]_ M )
16		sbceq1a 3342	. . . . . . . 8 |- ( x = X -> ( ph <-> [. X / x ]. ph ) )
17	15, 16	rabeqbidv 3108	. . . . . . 7 |- ( x = X -> { y e. [_ x / m ]_ M | ph } = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
18	10, 14, 17, 4	fvmptf 5966	. . . . . 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 661	. . . . 5 |- ( X e. dom F -> ( F ` X ) = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
20	19	eleq2d 2537	. . . 4 |- ( X e. dom F -> ( Y e. ( F ` X ) <-> Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } ) )
21		elrabi 3258	. . . . . 6 |- ( Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } -> Y e. [_ X / m ]_ M )
22	6, 21	anim12i 566	. . . . 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 434	. . . 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 215	. . 3 |- ( X e. dom F -> ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) ) )
25	1, 3, 24	3syl 20	. 2 |- ( Y e. ( F ` X ) -> ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) ) )
26	25	pm2.43i 47	1 |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   {crab 2818   _Vcvv 3113   [.wsbc 3331   [_csb 3435   (/)c0 3785   |-> cmpt 4505   dom cdm 4999   ` cfv 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596

This theorem is referenced by:  elfvmptrab  5971