Theorem elfvmptrab1 5986

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 3773	. . 3 |- ( Y e. ( F ` X ) -> ( F ` X ) =/= (/) )
2		ndmfv 5905	. . . 4 |- ( -. X e. dom F -> ( F ` X ) = (/) )
3	2	necon1ai 2662	. . 3 |- ( ( F ` X ) =/= (/) -> X e. dom F )
4		elfvmptrab1.f	. . . . . . . 8 |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
5	4	dmmptss 5351	. . . . . . 7 |- dom F C_ V
6	5	sseli 3466	. . . . . 6 |- ( X e. dom F -> X e. V )
7		elfvmptrab1.v	. . . . . . 7 |- ( X e. V -> [_ X / m ]_ M e. _V )
8		rabexg 4575	. . . . . . 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 2591	. . . . . . 7 |- F/_ x X
11		nfsbc1v 3325	. . . . . . . 8 |- F/ x [. X / x ]. ph
12		nfcv 2591	. . . . . . . . 9 |- F/_ x M
13	10, 12	nfcsb 3419	. . . . . . . 8 |- F/_ x [_ X / m ]_ M
14	11, 13	nfrab 3017	. . . . . . 7 |- F/_ x { y e. [_ X / m ]_ M | [. X / x ]. ph }
15		csbeq1 3404	. . . . . . . 8 |- ( x = X -> [_ x / m ]_ M = [_ X / m ]_ M )
16		sbceq1a 3316	. . . . . . . 8 |- ( x = X -> ( ph <-> [. X / x ]. ph ) )
17	15, 16	rabeqbidv 3082	. . . . . . 7 |- ( x = X -> { y e. [_ x / m ]_ M | ph } = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
18	10, 14, 17, 4	fvmptf 5982	. . . . . 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 665	. . . . 5 |- ( X e. dom F -> ( F ` X ) = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
20	19	eleq2d 2499	. . . 4 |- ( X e. dom F -> ( Y e. ( F ` X ) <-> Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } ) )
21		elrabi 3232	. . . . . 6 |- ( Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } -> Y e. [_ X / m ]_ M )
22	6, 21	anim12i 568	. . . . 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 435	. . . 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 218	. . 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 49	1 |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )

Syntax hints:   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1870   =/= wne 2625   {crab 2786   _Vcvv 3087   [.wsbc 3305   [_csb 3401   (/)c0 3767   |-> cmpt 4484   dom cdm 4854   ` cfv 5601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609

This theorem is referenced by:  elfvmptrab  5987