Theorem ovmpt3rab1 6891

Description: The value of an operation defined by the maps-to notation with a function into a class abstraction as a result. The domain of the function and the base set of the class abstraction may depend on the operands, using implicit substitution. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)

Hypotheses

Ref	Expression
ovmpt3rab1.o	|- O = ( x e. _V , y e. _V |-> ( z e. M |-> { a e. N | ph } ) )
ovmpt3rab1.m	|- ( ( x = X /\ y = Y ) -> M = K )
ovmpt3rab1.n	|- ( ( x = X /\ y = Y ) -> N = L )
ovmpt3rab1.p	|- ( ( x = X /\ y = Y ) -> ( ph <-> ps ) )
ovmpt3rab1.x	|- F/ x ps
ovmpt3rab1.y	|- F/ y ps

Assertion

Ref	Expression
ovmpt3rab1	|- ( ( X e. V /\ Y e. W /\ K e. U ) -> ( X O Y ) = ( z e. K |-> { a e. L | ps } ) )

Distinct variable groups:   x, K, y, z   L, a, x, y   N, a   x, V, y   x, W, y   x, U, y   X, a, x, y, z   Y, a, x, y, z

Allowed substitution hints:   ph( x, y, z, a)   ps( x, y, z, a)   U( z, a)   K( a)   L( z)   M( x, y, z, a)   N( x, y, z)   O( x, y, z, a)   V( z, a)   W( z, a)

Proof of Theorem ovmpt3rab1

Step	Hyp	Ref	Expression
1		ovmpt3rab1.o	. . 3 |- O = ( x e. _V , y e. _V |-> ( z e. M |-> { a e. N | ph } ) )
2	1	a1i 11	. 2 |- ( ( X e. V /\ Y e. W /\ K e. U ) -> O = ( x e. _V , y e. _V |-> ( z e. M |-> { a e. N | ph } ) ) )
3		ovmpt3rab1.m	. . . 4 |- ( ( x = X /\ y = Y ) -> M = K )
4		ovmpt3rab1.n	. . . . 5 |- ( ( x = X /\ y = Y ) -> N = L )
5		ovmpt3rab1.p	. . . . 5 |- ( ( x = X /\ y = Y ) -> ( ph <-> ps ) )
6	4, 5	rabeqbidv 3195	. . . 4 |- ( ( x = X /\ y = Y ) -> { a e. N | ph } = { a e. L | ps } )
7	3, 6	mpteq12dv 4733	. . 3 |- ( ( x = X /\ y = Y ) -> ( z e. M |-> { a e. N | ph } ) = ( z e. K |-> { a e. L | ps } ) )
8	7	adantl 482	. 2 |- ( ( ( X e. V /\ Y e. W /\ K e. U ) /\ ( x = X /\ y = Y ) ) -> ( z e. M |-> { a e. N | ph } ) = ( z e. K |-> { a e. L | ps } ) )
9		eqidd 2623	. 2 |- ( ( ( X e. V /\ Y e. W /\ K e. U ) /\ x = X ) -> _V = _V )
10		elex 3212	. . 3 |- ( X e. V -> X e. _V )
11	10	3ad2ant1 1082	. 2 |- ( ( X e. V /\ Y e. W /\ K e. U ) -> X e. _V )
12		elex 3212	. . 3 |- ( Y e. W -> Y e. _V )
13	12	3ad2ant2 1083	. 2 |- ( ( X e. V /\ Y e. W /\ K e. U ) -> Y e. _V )
14		mptexg 6484	. . 3 |- ( K e. U -> ( z e. K |-> { a e. L | ps } ) e. _V )
15	14	3ad2ant3 1084	. 2 |- ( ( X e. V /\ Y e. W /\ K e. U ) -> ( z e. K |-> { a e. L | ps } ) e. _V )
16		nfv 1843	. 2 |- F/ x ( X e. V /\ Y e. W /\ K e. U )
17		nfv 1843	. 2 |- F/ y ( X e. V /\ Y e. W /\ K e. U )
18		nfcv 2764	. 2 |- F/_ y X
19		nfcv 2764	. 2 |- F/_ x Y
20		nfcv 2764	. . 3 |- F/_ x K
21		ovmpt3rab1.x	. . . 4 |- F/ x ps
22		nfcv 2764	. . . 4 |- F/_ x L
23	21, 22	nfrab 3123	. . 3 |- F/_ x { a e. L | ps }
24	20, 23	nfmpt 4746	. 2 |- F/_ x ( z e. K |-> { a e. L | ps } )
25		nfcv 2764	. . 3 |- F/_ y K
26		ovmpt3rab1.y	. . . 4 |- F/ y ps
27		nfcv 2764	. . . 4 |- F/_ y L
28	26, 27	nfrab 3123	. . 3 |- F/_ y { a e. L | ps }
29	25, 28	nfmpt 4746	. 2 |- F/_ y ( z e. K |-> { a e. L | ps } )
30	2, 8, 9, 11, 13, 15, 16, 17, 18, 19, 24, 29	ovmpt2dxf 6786	1 |- ( ( X e. V /\ Y e. W /\ K e. U ) -> ( X O Y ) = ( z e. K |-> { a e. L | ps } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   {crab 2916   _Vcvv 3200   |-> cmpt 4729  (class class class)co 6650   |-> cmpt2 6652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  ovmpt3rabdm  6892  elovmpt3rab1  6893