Theorem elfvmptrab1 5656

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 5297	. . . 4 |- Fun F
3		funrel 5275	. . . 4 |- ( Fun F -> Rel F )
4	2, 3	ax-mp 5	. . 3 |- Rel F
5		relelfvdm 5590	. . 3 |- ( ( Rel F /\ Y e. ( F ` X ) ) -> X e. dom F )
6	4, 5	mpan 424	. 2 |- ( Y e. ( F ` X ) -> X e. dom F )
7	1	dmmptss 5166	. . . . . 6 |- dom F C_ V
8	7	sseli 3179	. . . . 5 |- ( X e. dom F -> X e. V )
9		elfvmptrab1.v	. . . . . 6 |- ( X e. V -> [_ X / m ]_ M e. _V )
10		rabexg 4176	. . . . . 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 2339	. . . . . 6 |- F/_ x X
13		nfsbc1v 3008	. . . . . . 7 |- F/ x [. X / x ]. ph
14		nfcv 2339	. . . . . . . 8 |- F/_ x M
15	12, 14	nfcsb 3122	. . . . . . 7 |- F/_ x [_ X / m ]_ M
16	13, 15	nfrabw 2678	. . . . . 6 |- F/_ x { y e. [_ X / m ]_ M | [. X / x ]. ph }
17		csbeq1 3087	. . . . . . 7 |- ( x = X -> [_ x / m ]_ M = [_ X / m ]_ M )
18		sbceq1a 2999	. . . . . . 7 |- ( x = X -> ( ph <-> [. X / x ]. ph ) )
19	17, 18	rabeqbidv 2758	. . . . . 6 |- ( x = X -> { y e. [_ x / m ]_ M | ph } = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
20	12, 16, 19, 1	fvmptf 5654	. . . . 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 411	. . . 4 |- ( X e. dom F -> ( F ` X ) = { y e. [_ X / m ]_ M | [. X / x ]. ph } )
22	21	eleq2d 2266	. . 3 |- ( X e. dom F -> ( Y e. ( F ` X ) <-> Y e. { y e. [_ X / m ]_ M | [. X / x ]. ph } ) )
23		elrabi 2917	. . . . 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 1364   e. wcel 2167   {crab 2479   _Vcvv 2763   [.wsbc 2989   [_csb 3084   |-> cmpt 4094   dom cdm 4663   Rel wrel 4668   Fun wfun 5252   ` cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fv 5266

This theorem is referenced by:  elfvmptrab  5657