Theorem elovmporab1w 6263

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 6257	. 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 3144	. . . . . . 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 3055	. . . . . . . 8 |- ( y = Y -> ( ph <-> [. Y / y ]. ph ) )
7		sbceq1a 3055	. . . . . . . 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 2810	. . . . 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 2235	. . . . 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 4260	. . . . . 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 2386	. . . . . . 7 |- F/_ x X
18	17	nfel1 2397	. . . . . 6 |- F/ x X e. _V
19		nfcv 2386	. . . . . . 7 |- F/_ x Y
20	19	nfel1 2397	. . . . . 6 |- F/ x Y e. _V
21	18, 20	nfan 1614	. . . . 5 |- F/ x ( X e. _V /\ Y e. _V )
22		nfcv 2386	. . . . . . 7 |- F/_ y X
23	22	nfel1 2397	. . . . . 6 |- F/ y X e. _V
24		nfcv 2386	. . . . . . 7 |- F/_ y Y
25	24	nfel1 2397	. . . . . 6 |- F/ y Y e. _V
26	23, 25	nfan 1614	. . . . 5 |- F/ y ( X e. _V /\ Y e. _V )
27		nfsbc1v 3064	. . . . . 6 |- F/ x [. X / x ]. [. Y / y ]. ph
28		nfcv 2386	. . . . . . 7 |- F/_ x M
29	17, 28	nfcsbw 3178	. . . . . 6 |- F/_ x [_ X / m ]_ M
30	27, 29	nfrabw 2727	. . . . 5 |- F/_ x { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph }
31		nfsbc1v 3064	. . . . . . 7 |- F/ y [. Y / y ]. ph
32	22, 31	nfsbcw 3176	. . . . . 6 |- F/ y [. X / x ]. [. Y / y ]. ph
33		nfcv 2386	. . . . . . 7 |- F/_ y M
34	22, 33	nfcsbw 3178	. . . . . 6 |- F/_ y [_ X / m ]_ M
35	32, 34	nfrabw 2727	. . . . 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 6187	. . . 4 |- ( ( X e. _V /\ Y e. _V ) -> ( X O Y ) = { z e. [_ X / m ]_ M | [. X / x ]. [. Y / y ]. ph } )
37	36	eleq2d 2304	. . 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 1007	. . . . 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 1490	. . . 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 2973	. . . 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 1005   = wceq 1398   e. wcel 2205   {crab 2526   _Vcvv 2815   [.wsbc 3045   [_csb 3141  (class class class)co 6058   e. cmpo 6060

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063

This theorem is referenced by:  elovmpowrd  11291