Theorem ovima0 6813

Description: An operation value is a member of the image plus null. (Contributed by Thierry Arnoux, 25-Jun-2019.)

Assertion

Ref	Expression
ovima0	|- ( ( X e. A /\ Y e. B ) -> ( X R Y ) e. ( ( R " ( A X. B ) ) u. { (/) } ) )

Proof of Theorem ovima0

Step	Hyp	Ref	Expression
1		simpr 477	. . 3 |- ( ( ( X e. A /\ Y e. B ) /\ ( X R Y ) = (/) ) -> ( X R Y ) = (/) )
2		ssun2 3777	. . . 4 |- { (/) } C_ ( ( R " ( A X. B ) ) u. { (/) } )
3		0ex 4790	. . . . 5 |- (/) e. _V
4	3	snid 4208	. . . 4 |- (/) e. { (/) }
5	2, 4	sselii 3600	. . 3 |- (/) e. ( ( R " ( A X. B ) ) u. { (/) } )
6	1, 5	syl6eqel 2709	. 2 |- ( ( ( X e. A /\ Y e. B ) /\ ( X R Y ) = (/) ) -> ( X R Y ) e. ( ( R " ( A X. B ) ) u. { (/) } ) )
7		ssun1 3776	. . 3 |- ( R " ( A X. B ) ) C_ ( ( R " ( A X. B ) ) u. { (/) } )
8		df-ov 6653	. . . 4 |- ( X R Y ) = ( R ` <. X , Y >. )
9		opelxpi 5148	. . . . 5 |- ( ( X e. A /\ Y e. B ) -> <. X , Y >. e. ( A X. B ) )
10	8	eqeq1i 2627	. . . . . . 7 |- ( ( X R Y ) = (/) <-> ( R ` <. X , Y >. ) = (/) )
11	10	notbii 310	. . . . . 6 |- ( -. ( X R Y ) = (/) <-> -. ( R ` <. X , Y >. ) = (/) )
12	11	biimpi 206	. . . . 5 |- ( -. ( X R Y ) = (/) -> -. ( R ` <. X , Y >. ) = (/) )
13		eliman0 6223	. . . . 5 |- ( ( <. X , Y >. e. ( A X. B ) /\ -. ( R ` <. X , Y >. ) = (/) ) -> ( R ` <. X , Y >. ) e. ( R " ( A X. B ) ) )
14	9, 12, 13	syl2an 494	. . . 4 |- ( ( ( X e. A /\ Y e. B ) /\ -. ( X R Y ) = (/) ) -> ( R ` <. X , Y >. ) e. ( R " ( A X. B ) ) )
15	8, 14	syl5eqel 2705	. . 3 |- ( ( ( X e. A /\ Y e. B ) /\ -. ( X R Y ) = (/) ) -> ( X R Y ) e. ( R " ( A X. B ) ) )
16	7, 15	sseldi 3601	. 2 |- ( ( ( X e. A /\ Y e. B ) /\ -. ( X R Y ) = (/) ) -> ( X R Y ) e. ( ( R " ( A X. B ) ) u. { (/) } ) )
17	6, 16	pm2.61dan 832	1 |- ( ( X e. A /\ Y e. B ) -> ( X R Y ) e. ( ( R " ( A X. B ) ) u. { (/) } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   (/)c0 3915   {csn 4177   <.cop 4183   X. cxp 5112   "cima 5117   ` cfv 5888  (class class class)co 6650

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-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-rab 2921  df-v 3202  df-sbc 3436  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-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  legval  25479