Theorem elfvmptrab1 5515

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		elfvmptrab1.f	. . . . 5 |- F = ( x e. V |-> { y e. [_ x / m ]_ M | ph } )
2	1	funmpt2 5162	. . . 4 |- Fun F
3		funrel 5140	. . . 4 |- ( Fun F -> Rel F )
4	2, 3	ax-mp 5	. . 3 |- Rel F
5		relelfvdm 5453	. . 3 |- ( ( Rel F /\ Y e. ( F ` X ) ) -> X e. dom F )
6	4, 5	mpan 420	. 2 |- ( Y e. ( F ` X ) -> X e. dom F )
7	1	dmmptss 5035	. . . . . 6 |- dom F C_ V
8	7	sseli 3093	. . . . 5 |- ( X e. dom F -> X e. V )
9		elfvmptrab1.v	. . . . . 6 |- ( X e. V -> [_ X / m ]_ M e. _V )
10		rabexg 4071	. . . . . 6 |- ( [_ X / m ]_ M e. _V -> { y e. [_ X / m ]_ M | [. X / x ]. ph } e. _V )
11	8, 9, 10	3syl 17	. . . . 5 |- ( X e. dom F -> { y e. [_ X / m ]_ M | [. X / x ]. ph } e. _V )
12		nfcv 2281	. . . . . 6 |- F/_ x X
13		nfsbc1v 2927	. . . . . . 7 |- F/ x [. X / x ]. ph
14		nfcv 2281	. . . . . . . 8 |- F/_ x M
15	12, 14	nfcsb 3037	. . . . . . 7 |- F/_ x [_ X / m ]_ M
16	13, 15	nfrabxy 2611	. . . . . 6 |- F/_ x { y e. [_ X / m ]_ M | [. X / x ]. ph }
17		csbeq1 3006	. . . . . . 7 |- ( x = X -> [_ x / m ]_ M = [_ X / m ]_ M )
18		sbceq1a 2918	. . . . . . 7 |- ( x = X -> ( ph <-> [. X / x ]. ph ) )
19	17, 18	rabeqbidv 2681	. . . . . 6 |- ( x = X -> { y e. [_ x / m ]_ M | ph } = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
20	12, 16, 19, 1	fvmptf 5513	. . . . 5 |- ( ( X e. V /\ { y e. [_ X / m ]_ M | [. X / x ]. ph } e. _V ) -> ( F ` X ) = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
21	8, 11, 20	syl2anc 408	. . . 4 |- ( X e. dom F -> ( F ` X ) = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
22	21	eleq2d 2209	. . 3 |- ( X e. dom F -> ( Y e. ( F ` X ) <-> Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } ) )
23		elrabi 2837	. . . . 5 |- ( Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } -> Y e. [_ X / m ]_ M )
24	8, 23	anim12i 336	. . . 4 |- ( ( X e. dom F /\ Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )
25	24	ex 114	. . 3 |- ( X e. dom F -> ( Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } -> ( X e. V /\ Y e. [_ X / m ]_ M ) ) )
26	22, 25	sylbid 149	. 2 |- ( X e. dom F -> ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) ) )
27	6, 26	mpcom 36	1 |- ( Y e. ( F ` X ) -> ( X e. V /\ Y e. [_ X / m ]_ M ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   {crab 2420   _Vcvv 2686   [.wsbc 2909   [_csb 3003   |-> cmpt 3989   dom cdm 4539   Rel wrel 4544   Fun wfun 5117   ` cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fv 5131

This theorem is referenced by:  elfvmptrab  5516