Theorem elovmpt2rab 6503

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
elovmpt2rab.o	|- O = ( x e. _V , y e. _V |-> { z e. M | ph } )
elovmpt2rab.v	|- ( ( X e. _V /\ Y e. _V ) -> M e. _V )

Assertion

Ref	Expression
elovmpt2rab	|- ( 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 elovmpt2rab

Step	Hyp	Ref	Expression
1		elovmpt2rab.o	. . 3 |- O = ( x e. _V , y e. _V |-> { z e. M | ph } )
2	1	elmpt2cl 6499	. 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. M | ph } ) )
4		sbceq1a 3342	. . . . . . . 8 |- ( y = Y -> ( ph <-> [. Y / y ]. ph ) )
5		sbceq1a 3342	. . . . . . . 8 |- ( x = X -> ( [. Y / y ]. ph <-> [. X / x ]. [. Y / y ]. ph ) )
6	4, 5	sylan9bbr 700	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( ph <-> [. X / x ]. [. Y / y ]. ph ) )
7	6	adantl 466	. . . . . 6 |- ( ( ( X e. _V /\ Y e. _V ) /\ ( x = X /\ y = Y ) ) -> ( ph <-> [. X / x ]. [. Y / y ]. ph ) )
8	7	rabbidv 3105	. . . . 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 2468	. . . . 5 |- ( ( ( X e. _V /\ Y e. _V ) /\ x = X ) -> _V = _V )
10		simpl 457	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> X e. _V )
11		simpr 461	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> Y e. _V )
12		elovmpt2rab.v	. . . . . 6 |- ( ( X e. _V /\ Y e. _V ) -> M e. _V )
13		rabexg 4597	. . . . . 6 |- ( M e. _V -> { z e. M | [. X / x ]. [. Y / y ]. ph } e. _V )
14	12, 13	syl 16	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> { z e. M | [. X / x ]. [. Y / y ]. ph } e. _V )
15		nfcv 2629	. . . . . . 7 |- F/_ x X
16	15	nfel1 2645	. . . . . 6 |- F/ x X e. _V
17		nfcv 2629	. . . . . . 7 |- F/_ x Y
18	17	nfel1 2645	. . . . . 6 |- F/ x Y e. _V
19	16, 18	nfan 1875	. . . . 5 |- F/ x ( X e. _V /\ Y e. _V )
20		nfcv 2629	. . . . . . 7 |- F/_ y X
21	20	nfel1 2645	. . . . . 6 |- F/ y X e. _V
22		nfcv 2629	. . . . . . 7 |- F/_ y Y
23	22	nfel1 2645	. . . . . 6 |- F/ y Y e. _V
24	21, 23	nfan 1875	. . . . 5 |- F/ y ( X e. _V /\ Y e. _V )
25		nfsbc1v 3351	. . . . . 6 |- F/ x [. X / x ]. [. Y / y ]. ph
26		nfcv 2629	. . . . . 6 |- F/_ x M
27	25, 26	nfrab 3043	. . . . 5 |- F/_ x { z e. M | [. X / x ]. [. Y / y ]. ph }
28		nfsbc1v 3351	. . . . . . 7 |- F/ y [. Y / y ]. ph
29	20, 28	nfsbc 3353	. . . . . 6 |- F/ y [. X / x ]. [. Y / y ]. ph
30		nfcv 2629	. . . . . 6 |- F/_ y M
31	29, 30	nfrab 3043	. . . . 5 |- F/_ y { z e. M | [. X / x ]. [. Y / y ]. ph }
32	3, 8, 9, 10, 11, 14, 19, 24, 20, 17, 27, 31	ovmpt2dxf 6410	. . . 4 |- ( ( X e. _V /\ Y e. _V ) -> ( X O Y ) = { z e. M | [. X / x ]. [. Y / y ]. ph } )
33	32	eleq2d 2537	. . 3 |- ( ( X e. _V /\ Y e. _V ) -> ( Z e. ( X O Y ) <-> Z e. { z e. M | [. X / x ]. [. Y / y ]. ph } ) )
34		elrabi 3258	. . . . 5 |- ( Z e. { z e. M | [. X / x ]. [. Y / y ]. ph } -> Z e. M )
35		df-3an 975	. . . . . 6 |- ( ( X e. _V /\ Y e. _V /\ Z e. M ) <-> ( ( X e. _V /\ Y e. _V ) /\ Z e. M ) )
36	35	simplbi2com 627	. . . . 5 |- ( Z e. M -> ( ( X e. _V /\ Y e. _V ) -> ( X e. _V /\ Y e. _V /\ Z e. M ) ) )
37	34, 36	syl 16	. . . 4 |- ( Z e. { z e. M | [. X / x ]. [. Y / y ]. ph } -> ( ( X e. _V /\ Y e. _V ) -> ( X e. _V /\ Y e. _V /\ Z e. M ) ) )
38	37	com12 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 ) ) )
39	33, 38	sylbid 215	. 2 |- ( ( X e. _V /\ Y e. _V ) -> ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. M ) ) )
40	2, 39	mpcom 36	1 |- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. M ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   {crab 2818   _Vcvv 3113   [.wsbc 3331  (class class class)co 6282   |-> cmpt2 6284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287

This theorem is referenced by: (None)