Theorem elovmporab1w 6205

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.) (Revised by GG, 26-Jan-2024.)

Hypotheses

Ref	Expression
elovmporab1w.o	|- O = ( x e. _V , y e. _V |-> { z e. [_ x / m ]_ M | ph } )
elovmporab1w.v	|- ( ( X e. _V /\ Y e. _V ) -> [_ X / m ]_ M e. _V )

Assertion

Ref	Expression
elovmporab1w	|- ( 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   x, m, y, z

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 elovmporab1w

Step	Hyp	Ref	Expression
1		elovmporab1w.o	. . 3 |- O = ( x e. _V , y e. _V |-> { z e. [_ x / m ]_ M | ph } )
2	1	elmpocl 6199	. 2 |- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V ) )
3	1	a1i 9	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> O = ( x e. _V , y e. _V |-> { z e. [_ x / m ]_ M | ph } ) )
4		csbeq1 3127	. . . . . . 7 |- ( x = X -> [_ x / m ]_ M = [_ X / m ]_ M )
5	4	ad2antrl 490	. . . . . 6 |- ( ( ( X e. _V /\ Y e. _V ) /\ ( x = X /\ y = Y ) ) -> [_ x / m ]_ M = [_ X / m ]_ M )
6		sbceq1a 3038	. . . . . . . 8 |- ( y = Y -> ( ph <-> [. Y / y ]. ph ) )
7		sbceq1a 3038	. . . . . . . 8 |- ( x = X -> ( [. Y / y ]. ph <-> [. X / x ]. [. Y / y ]. ph ) )
8	6, 7	sylan9bbr 463	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( ph <-> [. X / x ]. [. Y / y ]. ph ) )
9	8	adantl 277	. . . . . 6 |- ( ( ( X e. _V /\ Y e. _V ) /\ ( x = X /\ y = Y ) ) -> ( ph <-> [. X / x ]. [. Y / y ]. ph ) )
10	5, 9	rabeqbidv 2794	. . . . 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 2230	. . . . 5 |- ( ( ( X e. _V /\ Y e. _V ) /\ x = X ) -> _V = _V )
12		simpl 109	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> X e. _V )
13		simpr 110	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> Y e. _V )
14		elovmporab1w.v	. . . . . 6 |- ( ( X e. _V /\ Y e. _V ) -> [_ X / m ]_ M e. _V )
15		rabexg 4226	. . . . . 6 |- ( [_ X / m ]_ M e. _V -> { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } e. _V )
16	14, 15	syl 14	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } e. _V )
17		nfcv 2372	. . . . . . 7 |- F/_ x X
18	17	nfel1 2383	. . . . . 6 |- F/ x X e. _V
19		nfcv 2372	. . . . . . 7 |- F/_ x Y
20	19	nfel1 2383	. . . . . 6 |- F/ x Y e. _V
21	18, 20	nfan 1611	. . . . 5 |- F/ x ( X e. _V /\ Y e. _V )
22		nfcv 2372	. . . . . . 7 |- F/_ y X
23	22	nfel1 2383	. . . . . 6 |- F/ y X e. _V
24		nfcv 2372	. . . . . . 7 |- F/_ y Y
25	24	nfel1 2383	. . . . . 6 |- F/ y Y e. _V
26	23, 25	nfan 1611	. . . . 5 |- F/ y ( X e. _V /\ Y e. _V )
27		nfsbc1v 3047	. . . . . 6 |- F/ x [. X / x ]. [. Y / y ]. ph
28		nfcv 2372	. . . . . . 7 |- F/_ x M
29	17, 28	nfcsbw 3161	. . . . . 6 |- F/_ x [_ X / m ]_ M
30	27, 29	nfrabw 2712	. . . . 5 |- F/_ x { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph }
31		nfsbc1v 3047	. . . . . . 7 |- F/ y [. Y / y ]. ph
32	22, 31	nfsbcw 3159	. . . . . 6 |- F/ y [. X / x ]. [. Y / y ]. ph
33		nfcv 2372	. . . . . . 7 |- F/_ y M
34	22, 33	nfcsbw 3161	. . . . . 6 |- F/_ y [_ X / m ]_ M
35	32, 34	nfrabw 2712	. . . . 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	ovmpodxf 6129	. . . 4 |- ( ( X e. _V /\ Y e. _V ) -> ( X O Y ) = { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } )
37	36	eleq2d 2299	. . 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 1004	. . . . 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 1487	. . . 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 2956	. . . 4 |- ( Z e. { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } -> Z e. [_ X / m ]_ M )
41	39, 40	syl11 31	. . 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 150	. 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 36	1 |- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. [_ X / m ]_ M ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   [.wsbc 3028   [_csb 3124  (class class class)co 6000   e. cmpo 6002

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005

This theorem is referenced by:  elovmpowrd  11108