Theorem elovmporab 6232

Description: Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)

Hypotheses

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

Assertion

Ref	Expression
elovmporab	|- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. M ) )

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

Allowed substitution hints:   ph( x, y, z)   O( x, y, z)   Z( x, y)

Proof of Theorem elovmporab

Step	Hyp	Ref	Expression
1		elovmporab.o	. . 3 |- O = ( x e. _V , y e. _V |-> { z e. M | ph } )
2	1	elmpocl 6227	. 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. M | ph } ) )
4		sbceq1a 3042	. . . . . . . 8 |- ( y = Y -> ( ph <-> [. Y / y ]. ph ) )
5		sbceq1a 3042	. . . . . . . 8 |- ( x = X -> ( [. Y / y ]. ph <-> [. X / x ]. [. Y / y ]. ph ) )
6	4, 5	sylan9bbr 463	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( ph <-> [. X / x ]. [. Y / y ]. ph ) )
7	6	adantl 277	. . . . . 6 |- ( ( ( X e. _V /\ Y e. _V ) /\ ( x = X /\ y = Y ) ) -> ( ph <-> [. X / x ]. [. Y / y ]. ph ) )
8	7	rabbidv 2792	. . . . 5 |- ( ( ( X e. _V /\ Y e. _V ) /\ ( x = X /\ y = Y ) ) -> { z e. M | ph } = { z e. M | [. X / x ]. [. Y / y ]. ph } )
9		eqidd 2232	. . . . 5 |- ( ( ( X e. _V /\ Y e. _V ) /\ x = X ) -> _V = _V )
10		simpl 109	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> X e. _V )
11		simpr 110	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> Y e. _V )
12		elovmporab.v	. . . . . 6 |- ( ( X e. _V /\ Y e. _V ) -> M e. _V )
13		rabexg 4238	. . . . . 6 |- ( M e. _V -> { z e. M | [. X / x ]. [. Y / y ]. ph } e. _V )
14	12, 13	syl 14	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> { z e. M | [. X / x ]. [. Y / y ]. ph } e. _V )
15		nfcv 2375	. . . . . . 7 |- F/_ x X
16	15	nfel1 2386	. . . . . 6 |- F/ x X e. _V
17		nfcv 2375	. . . . . . 7 |- F/_ x Y
18	17	nfel1 2386	. . . . . 6 |- F/ x Y e. _V
19	16, 18	nfan 1614	. . . . 5 |- F/ x ( X e. _V /\ Y e. _V )
20		nfcv 2375	. . . . . . 7 |- F/_ y X
21	20	nfel1 2386	. . . . . 6 |- F/ y X e. _V
22		nfcv 2375	. . . . . . 7 |- F/_ y Y
23	22	nfel1 2386	. . . . . 6 |- F/ y Y e. _V
24	21, 23	nfan 1614	. . . . 5 |- F/ y ( X e. _V /\ Y e. _V )
25		nfsbc1v 3051	. . . . . 6 |- F/ x [. X / x ]. [. Y / y ]. ph
26		nfcv 2375	. . . . . 6 |- F/_ x M
27	25, 26	nfrabw 2715	. . . . 5 |- F/_ x { z e. M | [. X / x ]. [. Y / y ]. ph }
28		nfsbc1v 3051	. . . . . . 7 |- F/ y [. Y / y ]. ph
29	20, 28	nfsbcw 3163	. . . . . 6 |- F/ y [. X / x ]. [. Y / y ]. ph
30		nfcv 2375	. . . . . 6 |- F/_ y M
31	29, 30	nfrabw 2715	. . . . 5 |- F/_ y { z e. M | [. X / x ]. [. Y / y ]. ph }
32	3, 8, 9, 10, 11, 14, 19, 24, 20, 17, 27, 31	ovmpodxf 6157	. . . 4 |- ( ( X e. _V /\ Y e. _V ) -> ( X O Y ) = { z e. M | [. X / x ]. [. Y / y ]. ph } )
33	32	eleq2d 2301	. . 3 |- ( ( X e. _V /\ Y e. _V ) -> ( Z e. ( X O Y ) <-> Z e. { z e. M | [. X / x ]. [. Y / y ]. ph } ) )
34		df-3an 1007	. . . . 5 |- ( ( X e. _V /\ Y e. _V /\ Z e. M ) <-> ( ( X e. _V /\ Y e. _V ) /\ Z e. M ) )
35	34	simplbi2com 1490	. . . 4 |- ( Z e. M -> ( ( X e. _V /\ Y e. _V ) -> ( X e. _V /\ Y e. _V /\ Z e. M ) ) )
36		elrabi 2960	. . . 4 |- ( Z e. { z e. M | [. X / x ]. [. Y / y ]. ph } -> Z e. M )
37	35, 36	syl11 31	. . 3 |- ( ( X e. _V /\ Y e. _V ) -> ( Z e. { z e. M | [. X / x ]. [. Y / y ]. ph } -> ( X e. _V /\ Y e. _V /\ Z e. M ) ) )
38	33, 37	sylbid 150	. 2 |- ( ( X e. _V /\ Y e. _V ) -> ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. M ) ) )
39	2, 38	mpcom 36	1 |- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. M ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   {crab 2515   _Vcvv 2803   [.wsbc 3032  (class class class)co 6028   e. cmpo 6030

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033

This theorem is referenced by: (None)