Theorem elovmpt3imp 6890

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 must be sets. Remark: a function which is the result of an operation can be regared as operation with 3 operands - therefore the abbreviation "mpt3" is used in the label. (Contributed by AV, 16-May-2019.)

Hypothesis

Ref	Expression
elovmpt3imp.o	|- O = ( x e. _V , y e. _V |-> ( z e. M |-> B ) )

Assertion

Ref	Expression
elovmpt3imp	|- ( A e. ( ( X O Y ) ` Z ) -> ( X e. _V /\ Y e. _V ) )

Distinct variable group:   x, y

Allowed substitution hints:   A( x, y, z)   B( x, y, z)   M( x, y, z)   O( x, y, z)   X( x, y, z)   Y( x, y, z)   Z( x, y, z)

Proof of Theorem elovmpt3imp

Step	Hyp	Ref	Expression
1		ne0i 3921	. 2 |- ( A e. ( ( X O Y ) ` Z ) -> ( ( X O Y ) ` Z ) =/= (/) )
2		ax-1 6	. . 3 |- ( ( X e. _V /\ Y e. _V ) -> ( ( ( X O Y ) ` Z ) =/= (/) -> ( X e. _V /\ Y e. _V ) ) )
3		elovmpt3imp.o	. . . . 5 |- O = ( x e. _V , y e. _V |-> ( z e. M |-> B ) )
4	3	mpt2ndm0 6875	. . . 4 |- ( -. ( X e. _V /\ Y e. _V ) -> ( X O Y ) = (/) )
5		fveq1 6190	. . . . 5 |- ( ( X O Y ) = (/) -> ( ( X O Y ) ` Z ) = ( (/) ` Z ) )
6		0fv 6227	. . . . 5 |- ( (/) ` Z ) = (/)
7	5, 6	syl6eq 2672	. . . 4 |- ( ( X O Y ) = (/) -> ( ( X O Y ) ` Z ) = (/) )
8		eqneqall 2805	. . . 4 |- ( ( ( X O Y ) ` Z ) = (/) -> ( ( ( X O Y ) ` Z ) =/= (/) -> ( X e. _V /\ Y e. _V ) ) )
9	4, 7, 8	3syl 18	. . 3 |- ( -. ( X e. _V /\ Y e. _V ) -> ( ( ( X O Y ) ` Z ) =/= (/) -> ( X e. _V /\ Y e. _V ) ) )
10	2, 9	pm2.61i 176	. 2 |- ( ( ( X O Y ) ` Z ) =/= (/) -> ( X e. _V /\ Y e. _V ) )
11	1, 10	syl 17	1 |- ( A e. ( ( X O Y ) ` Z ) -> ( X e. _V /\ Y e. _V ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   ` cfv 5888  (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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  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-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-xp 5120  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  elovmpt3rab1  6893  elovmptnn0wrd  13348