Theorem ovmpt3rabdm 6892

Description: If the value of a function which is the result of an operation defined by the maps-to notation is not empty, the operands and the argument of the function must be sets. (Contributed 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 )

Assertion

Ref	Expression
ovmpt3rabdm	|- ( ( ( X e. V /\ Y e. W /\ K e. U ) /\ L e. T ) -> dom ( X O Y ) = K )

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   z, L   z, T   z, U   z, V   z, W

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

Proof of Theorem ovmpt3rabdm

Step	Hyp	Ref	Expression
1		ovmpt3rab1.o	. . . . 5 |- O = ( x e. _V , y e. _V |-> ( z e. M |-> { a e. N | ph } ) )
2		ovmpt3rab1.m	. . . . 5 |- ( ( x = X /\ y = Y ) -> M = K )
3		ovmpt3rab1.n	. . . . 5 |- ( ( x = X /\ y = Y ) -> N = L )
4		sbceq1a 3446	. . . . . 6 |- ( y = Y -> ( ph <-> [. Y / y ]. ph ) )
5		sbceq1a 3446	. . . . . 6 |- ( x = X -> ( [. Y / y ]. ph <-> [. X / x ]. [. Y / y ]. ph ) )
6	4, 5	sylan9bbr 737	. . . . 5 |- ( ( x = X /\ y = Y ) -> ( ph <-> [. X / x ]. [. Y / y ]. ph ) )
7		nfsbc1v 3455	. . . . 5 |- F/ x [. X / x ]. [. Y / y ]. ph
8		nfcv 2764	. . . . . 6 |- F/_ y X
9		nfsbc1v 3455	. . . . . 6 |- F/ y [. Y / y ]. ph
10	8, 9	nfsbc 3457	. . . . 5 |- F/ y [. X / x ]. [. Y / y ]. ph
11	1, 2, 3, 6, 7, 10	ovmpt3rab1 6891	. . . 4 |- ( ( X e. V /\ Y e. W /\ K e. U ) -> ( X O Y ) = ( z e. K |-> { a e. L | [. X / x ]. [. Y / y ]. ph } ) )
12	11	adantr 481	. . 3 |- ( ( ( X e. V /\ Y e. W /\ K e. U ) /\ L e. T ) -> ( X O Y ) = ( z e. K |-> { a e. L | [. X / x ]. [. Y / y ]. ph } ) )
13	12	dmeqd 5326	. 2 |- ( ( ( X e. V /\ Y e. W /\ K e. U ) /\ L e. T ) -> dom ( X O Y ) = dom ( z e. K |-> { a e. L | [. X / x ]. [. Y / y ]. ph } ) )
14		rabexg 4812	. . . . 5 |- ( L e. T -> { a e. L | [. X / x ]. [. Y / y ]. ph } e. _V )
15	14	adantl 482	. . . 4 |- ( ( ( X e. V /\ Y e. W /\ K e. U ) /\ L e. T ) -> { a e. L | [. X / x ]. [. Y / y ]. ph } e. _V )
16	15	ralrimivw 2967	. . 3 |- ( ( ( X e. V /\ Y e. W /\ K e. U ) /\ L e. T ) -> A. z e. K { a e. L | [. X / x ]. [. Y / y ]. ph } e. _V )
17		dmmptg 5632	. . 3 |- ( A. z e. K { a e. L | [. X / x ]. [. Y / y ]. ph } e. _V -> dom ( z e. K |-> { a e. L | [. X / x ]. [. Y / y ]. ph } ) = K )
18	16, 17	syl 17	. 2 |- ( ( ( X e. V /\ Y e. W /\ K e. U ) /\ L e. T ) -> dom ( z e. K |-> { a e. L | [. X / x ]. [. Y / y ]. ph } ) = K )
19	13, 18	eqtrd 2656	1 |- ( ( ( X e. V /\ Y e. W /\ K e. U ) /\ L e. T ) -> dom ( X O Y ) = K )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   [.wsbc 3435   |-> cmpt 4729   dom cdm 5114  (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:  elovmpt3rab1  6893