Theorem elovmpt2rab 6534

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 6530	. 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 3266	. . . . . . . 8 |- ( y = Y -> ( ph <-> [. Y / y ]. ph ) )
5		sbceq1a 3266	. . . . . . . 8 |- ( x = X -> ( [. Y / y ]. ph <-> [. X / x ]. [. Y / y ]. ph ) )
6	4, 5	sylan9bbr 715	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( ph <-> [. X / x ]. [. Y / y ]. ph ) )
7	6	adantl 473	. . . . . 6 |- ( ( ( X e. _V /\ Y e. _V ) /\ ( x = X /\ y = Y ) ) -> ( ph <-> [. X / x ]. [. Y / y ]. ph ) )
8	7	rabbidv 3022	. . . . 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 2472	. . . . 5 |- ( ( ( X e. _V /\ Y e. _V ) /\ x = X ) -> _V = _V )
10		simpl 464	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> X e. _V )
11		simpr 468	. . . . 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 4549	. . . . . 6 |- ( M e. _V -> { z e. M | [. X / x ]. [. Y / y ]. ph } e. _V )
14	12, 13	syl 17	. . . . 5 |- ( ( X e. _V /\ Y e. _V ) -> { z e. M | [. X / x ]. [. Y / y ]. ph } e. _V )
15		nfcv 2612	. . . . . . 7 |- F/_ x X
16	15	nfel1 2626	. . . . . 6 |- F/ x X e. _V
17		nfcv 2612	. . . . . . 7 |- F/_ x Y
18	17	nfel1 2626	. . . . . 6 |- F/ x Y e. _V
19	16, 18	nfan 2031	. . . . 5 |- F/ x ( X e. _V /\ Y e. _V )
20		nfcv 2612	. . . . . . 7 |- F/_ y X
21	20	nfel1 2626	. . . . . 6 |- F/ y X e. _V
22		nfcv 2612	. . . . . . 7 |- F/_ y Y
23	22	nfel1 2626	. . . . . 6 |- F/ y Y e. _V
24	21, 23	nfan 2031	. . . . 5 |- F/ y ( X e. _V /\ Y e. _V )
25		nfsbc1v 3275	. . . . . 6 |- F/ x [. X / x ]. [. Y / y ]. ph
26		nfcv 2612	. . . . . 6 |- F/_ x M
27	25, 26	nfrab 2958	. . . . 5 |- F/_ x { z e. M | [. X / x ]. [. Y / y ]. ph }
28		nfsbc1v 3275	. . . . . . 7 |- F/ y [. Y / y ]. ph
29	20, 28	nfsbc 3277	. . . . . 6 |- F/ y [. X / x ]. [. Y / y ]. ph
30		nfcv 2612	. . . . . 6 |- F/_ y M
31	29, 30	nfrab 2958	. . . . 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 6441	. . . 4 |- ( ( X e. _V /\ Y e. _V ) -> ( X O Y ) = { z e. M | [. X / x ]. [. Y / y ]. ph } )
33	32	eleq2d 2534	. . 3 |- ( ( X e. _V /\ Y e. _V ) -> ( Z e. ( X O Y ) <-> Z e. { z e. M | [. X / x ]. [. Y / y ]. ph } ) )
34		elrabi 3181	. . . . 5 |- ( Z e. { z e. M | [. X / x ]. [. Y / y ]. ph } -> Z e. M )
35		df-3an 1009	. . . . . 6 |- ( ( X e. _V /\ Y e. _V /\ Z e. M ) <-> ( ( X e. _V /\ Y e. _V ) /\ Z e. M ) )
36	35	simplbi2com 639	. . . . 5 |- ( Z e. M -> ( ( X e. _V /\ Y e. _V ) -> ( X e. _V /\ Y e. _V /\ Z e. M ) ) )
37	34, 36	syl 17	. . . 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 223	. 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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   {crab 2760   _Vcvv 3031   [.wsbc 3255  (class class class)co 6308   |-> cmpt2 6310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313

This theorem is referenced by: (None)