Theorem elovmpt2rab1 6881

Description: Implications for the value of an operation, 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 first operand. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Hypotheses

Ref	Expression
elovmpt2rab1.o	|- O = ( x e. _V , y e. _V |-> { z e. [_ x / m ]_ M | ph } )
elovmpt2rab1.v	|- ( ( X e. _V /\ Y e. _V ) -> [_ X / m ]_ M e. _V )

Assertion

Ref	Expression
elovmpt2rab1	|- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. [_ X / m ]_ M ) )

Distinct variable groups:   x, M, y, z   x, X, y, z   x, Y, y, z   z, Z   z, m

Allowed substitution hints:   ph( x, y, z, m)   M( m)   O( x, y, z, m)   X( m)   Y( m)   Z( x, y, m)

Proof of Theorem elovmpt2rab1

Step	Hyp	Ref	Expression
1		elovmpt2rab1.o	. . 3 |- O = ( x e. _V , y e. _V |-> { z e. [_ x / m ]_ M | ph } )
2	1	elmpt2cl 6876	. 2 |- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V ) )
3	1	a1i 11	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> O = ( x e. _V , y e. _V |-> { z e. [_ x / m ]_ M | ph } ) )
4		csbeq1 3536	. . . . . . 7 |- ( x = X -> [_ x / m ]_ M = [_ X / m ]_ M )
5	4	ad2antrl 764	. . . . . 6 |- ( ( ( X e. _V /\ Y e. _V ) /\ ( x = X /\ y = Y ) ) -> [_ x / m ]_ M = [_ X / m ]_ M )
6		sbceq1a 3446	. . . . . . . 8 |- ( y = Y -> ( ph <-> [. Y / y ]. ph ) )
7		sbceq1a 3446	. . . . . . . 8 |- ( x = X -> ( [. Y / y ]. ph <-> [. X / x ]. [. Y / y ]. ph ) )
8	6, 7	sylan9bbr 737	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( ph <-> [. X / x ]. [. Y / y ]. ph ) )
9	8	adantl 482	. . . . . 6 |- ( ( ( X e. _V /\ Y e. _V ) /\ ( x = X /\ y = Y ) ) -> ( ph <-> [. X / x ]. [. Y / y ]. ph ) )
10	5, 9	rabeqbidv 3195	. . . . 5 |- ( ( ( X e. _V /\ Y e. _V ) /\ ( x = X /\ y = Y ) ) -> { z e. [_ x / m ]_ M | ph } = { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } )
11		eqidd 2623	. . . . 5 |- ( ( ( X e. _V /\ Y e. _V ) /\ x = X ) -> _V = _V )
12		simpl 473	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> X e. _V )
13		simpr 477	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> Y e. _V )
14		elovmpt2rab1.v	. . . . . 6 |- ( ( X e. _V /\ Y e. _V ) -> [_ X / m ]_ M e. _V )
15		rabexg 4812	. . . . . 6 |- ( [_ X / m ]_ M e. _V -> { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } e. _V )
16	14, 15	syl 17	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } e. _V )
17		nfcv 2764	. . . . . . 7 |- F/_ x X
18	17	nfel1 2779	. . . . . 6 |- F/ x X e. _V
19		nfcv 2764	. . . . . . 7 |- F/_ x Y
20	19	nfel1 2779	. . . . . 6 |- F/ x Y e. _V
21	18, 20	nfan 1828	. . . . 5 |- F/ x ( X e. _V /\ Y e. _V )
22		nfcv 2764	. . . . . . 7 |- F/_ y X
23	22	nfel1 2779	. . . . . 6 |- F/ y X e. _V
24		nfcv 2764	. . . . . . 7 |- F/_ y Y
25	24	nfel1 2779	. . . . . 6 |- F/ y Y e. _V
26	23, 25	nfan 1828	. . . . 5 |- F/ y ( X e. _V /\ Y e. _V )
27		nfsbc1v 3455	. . . . . 6 |- F/ x [. X / x ]. [. Y / y ]. ph
28		nfcv 2764	. . . . . . 7 |- F/_ x M
29	17, 28	nfcsb 3551	. . . . . 6 |- F/_ x [_ X / m ]_ M
30	27, 29	nfrab 3123	. . . . 5 |- F/_ x { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph }
31		nfsbc1v 3455	. . . . . . 7 |- F/ y [. Y / y ]. ph
32	22, 31	nfsbc 3457	. . . . . 6 |- F/ y [. X / x ]. [. Y / y ]. ph
33		nfcv 2764	. . . . . . 7 |- F/_ y M
34	22, 33	nfcsb 3551	. . . . . 6 |- F/_ y [_ X / m ]_ M
35	32, 34	nfrab 3123	. . . . 5 |- F/_ y { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph }
36	3, 10, 11, 12, 13, 16, 21, 26, 22, 19, 30, 35	ovmpt2dxf 6786	. . . 4 |- ( ( X e. _V /\ Y e. _V ) -> ( X O Y ) = { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } )
37	36	eleq2d 2687	. . 3 |- ( ( X e. _V /\ Y e. _V ) -> ( Z e. ( X O Y ) <-> Z e. { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } ) )
38		df-3an 1039	. . . . 5 |- ( ( X e. _V /\ Y e. _V /\ Z e. [_ X / m ]_ M ) <-> ( ( X e. _V /\ Y e. _V ) /\ Z e. [_ X / m ]_ M ) )
39	38	simplbi2com 657	. . . 4 |- ( Z e. [_ X / m ]_ M -> ( ( X e. _V /\ Y e. _V ) -> ( X e. _V /\ Y e. _V /\ Z e. [_ X / m ]_ M ) ) )
40		elrabi 3359	. . . 4 |- ( Z e. { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } -> Z e. [_ X / m ]_ M )
41	39, 40	syl11 33	. . 3 |- ( ( X e. _V /\ Y e. _V ) -> ( Z e. { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } -> ( X e. _V /\ Y e. _V /\ Z e. [_ X / m ]_ M ) ) )
42	37, 41	sylbid 230	. 2 |- ( ( X e. _V /\ Y e. _V ) -> ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. [_ X / m ]_ M ) ) )
43	2, 42	mpcom 38	1 |- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. [_ X / m ]_ M ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   [.wsbc 3435   [_csb 3533  (class class class)co 6650   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  elovmpt2wrd  13347