Theorem elfvmptrab1 5794

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 5411	. . . 4 |- Fun F
3		funrel 5389	. . . 4 |- ( Fun F -> Rel F )
4	2, 3	ax-mp 5	. . 3 |- Rel F
5		relelfvdm 5722	. . 3 |- ( ( Rel F /\ Y e. ( F ` X ) ) -> X e. dom F )
6	4, 5	mpan 428	. 2 |- ( Y e. ( F ` X ) -> X e. dom F )
7	1	dmmptss 5279	. . . . . 6 |- dom F C_ V
8	7	sseli 3244	. . . . 5 |- ( X e. dom F -> X e. V )
9		elfvmptrab1.v	. . . . . 6 |- ( X e. V -> [_ X / m ]_ M e. _V )
10		rabexg 4274	. . . . . 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 2392	. . . . . 6 |- F/_ x X
13		nfsbc1v 3070	. . . . . . 7 |- F/ x [. X / x ]. ph
14		nfcv 2392	. . . . . . . 8 |- F/_ x M
15	12, 14	nfcsb 3185	. . . . . . 7 |- F/_ x [_ X / m ]_ M
16	13, 15	nfrabw 2733	. . . . . 6 |- F/_ x { y e. [_ X / m ]_ M | [. X / x ]. ph }
17		csbeq1 3150	. . . . . . 7 |- ( x = X -> [_ x / m ]_ M = [_ X / m ]_ M )
18		sbceq1a 3061	. . . . . . 7 |- ( x = X -> ( ph <-> [. X / x ]. ph ) )
19	17, 18	rabeqbidv 2816	. . . . . 6 |- ( x = X -> { y e. [_ x / m ]_ M | ph } = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
20	12, 16, 19, 1	fvmptf 5792	. . . . 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 415	. . . 4 |- ( X e. dom F -> ( F ` X ) = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
22	21	eleq2d 2308	. . 3 |- ( X e. dom F -> ( Y e. ( F ` X ) <-> Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } ) )
23		elrabi 2979	. . . . 5 |- ( Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } -> Y e. [_ X / m ]_ M )
24	8, 23	anim12i 338	. . . 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 115	. . 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 150	. 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 104   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   [.wsbc 3051   [_csb 3147   |-> cmpt 4187   dom cdm 4769   Rel wrel 4774   Fun wfun 5366   ` cfv 5372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fv 5380

This theorem is referenced by:  elfvmptrab  5795